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How to Use Filter Equations in Software and Spreadsheets

How to Use Filter Equations in Software and Spreadsheets - RF CafeDo a World Wide Web (WWW) search for filter equations and you will find thousands of pages, including a few here on RF Cafe. However, if you want an example of how to implement the transfer functions in a spreadsheet or software, examples of actual code are elusive (other than maybe a Matlab or MathCAD worksheet). As one who has incorporated equations for Butterworth, Chebyshev Type 1, Chebyshev Type 2, Bessel, and other filter functions in many spreadsheets and software over the past few decades, I figured it might be useful to post snippets of my code so that someone else can copy and paste it directly into other work. BTW, I do not consider myself to be a filter expert by any means and there is no ground-breaking knowledge here; it's just hopefully easier to find. Please look at my Filter Transfer Functions page for some of the textbook equations.

Writing a macro to use in a spreadsheet is the preferred approach for the sake of maintenance. Excel still supports Visual Basic for Applications (VBA), and since its code format is very readable, that is what is shown here. They can be adapted very easily for use in other languages. Since the VBA code below was copied and pasted directly from my Excel (2007) spreadsheet, you should be able to copy and paste it directly into yours.

The following definitions are used in all the equations:

f = frequency for calculation of gain    |    fL = lower cutoff frequency    |    fU = upper cutoff frequency

fRel = frequency relative to the cutoff frequency    |    BW = bandwidth    |    N = filter order*

Ripple = gain ripple (dB)    |    PBRipple = passband ripple (dB)    |    SBRipple = stopband ripple (dB)

*  Note that for calculation purposes N does not have to be an integer. That allows you to tweak the filter model to match the actual measured results. It will have a noticeable effect on the appearance of inband and out-of-band ripple spacing, but not min/max amplitude.

Excel formula cell references for filter macros - RF CafeMacro formula calls from within a spreadsheet cell look similar to this:

=Cheby1BPGain($B48, $C$8, $C$7, $C$16,$C$18) ,

where the Column/Row cell references are where the defined values of:

cell B48 = Frequency for which gain is being calculated

cell C8 = Lower cutoff frequency

cell C7 = Upper cutoff frequency

cell C16 = Filter order

cell C18 = Passband ripple (dB)

Filter function names are colored red in the declaration and everywhere they are assigned a value.

Since logarithmic charts are used, a check is made to assure negative and zero (0) values for frequency are not calculated, which could result in a calculation and/or graph scaling error.

Butterworth Filters | Chebyshev Type 1 Filters | Chebyshev Type 2 Filters | Bessel Filters


Butterworth Filters

Butterworth filters are probably the simplest to implement because the equations are the same both inside and outside the passband.

 

Public Function ButterLowpassGain(f As Double, fU As Double, N As Double) As Double

Dim fRel As Double

  fRel = f / fU

  ButterLowpassGain = -10 * WorksheetFunction.Log10(1 + (fRel) ^ (2 * N))

End Function

 

Public Function ButterHighpassGain(f As Double, fL As Double, N As Double) As Double

Dim fRel As Double

  fRel = fL / f

  ButterHighpassGain = -10 * WorksheetFunction.Log10(1 + (fRel) ^ (2 * N))

End Function

 

Public Function ButterBandpassGain(f As Double, fL As Double, fU As Double, N As Double) As Double

Dim fRel As Double

Dim fo As Double

Dim BW As Double

  fo = (fU * fL) ^ 0.5

  BW = (fU - fL) / fo

  fRel = ((f / fo) - (fo / f)) / BW

  ButterBandpassGain = -10 * WorksheetFunction.Log10(1 + (fRel) ^ (2 * N))

End Function

 

Public Function ButterBandstopGain(f As Double, fL As Double, fU As Double, N As Double) As Double

Dim fRel As Double

Dim fo As Double

Dim BW As Double

  fo = (fU * fL) ^ 0.5

  BW = (fU - fL) / fo

  fRel = 1 / (((f / fo) - (fo / f)) / BW)

  ButterBandstopGain = -10 * WorksheetFunction.Log10(1 + (fRel) ^ (2 * N))

End Function

 

Chebyshev Type 1 Filters

Chebyshev Type 1 filters have two distinct regions where the transfer function are different. The inband region is a standard cosine function whereas the out-of-band region is a hyperbolic cosine function. This requires checking to determine whether the frequency used for calculation is in-band or out-of-band. Note that the "esqr" term is different than that of the Chebyshev Type 2 filter.

 

Public Function Cheby1LPGain(f As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

  fRel = f / fU

  esqr = 10 ^ (Ripple / 10) - 1

  If fRel < 1 Then     'In-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else     'Out-of-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby1LPGain = -10 * WorksheetFunction.Log10(1 + esqr * Cn ^ 2)

End Function

 

Public Function Cheby1HPGain(f As Double, fL As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

  fRel = fL / f

  esqr = 10 ^ (Ripple / 10) - 1

  If fRel < 1 Then     'Out-of-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else     'In-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby1HPGain = -10 * WorksheetFunction.Log10(1 + esqr * Cn ^ 2)

End Function

 

Public Function Cheby1BPGain(f As Double, fL As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

Dim fo As Double

Dim BW As Double

  fo = Sqr(fU * fL)

  BW = (fU - fL) / fo

  fRel = Abs(((f / fo) - (fo / f)) / BW)

  esqr = 10 ^ (Ripple / 10) - 1

  If (f < fL) Or (f > fU) Then     'Out-of-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  Else     'In-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  End If

  Cheby1BPGain = -10 * WorksheetFunction.Log10(1 + esqr * Cn ^ 2)

End Function

 

Public Function Cheby1BSGain(f As Double, fL As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

Dim fo As Double

Dim BW As Double

  fo = Sqr(fU * fL)

  BW = (fU - fL) / fo

  fRel = 1 / Abs(((f / fo) - (fo / f)) / BW)

  esqr = 10 ^ (Ripple / 10) - 1

  If (f <= fL) Or (f >= fU) Then     'In-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else     'Out-of-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby1BSGain = -10 * WorksheetFunction.Log10(1 + esqr * Cn ^ 2)

End Function

 

Chebyshev Type 2 Filters

Chebyshev Type 2 filters have two distinct regions where the transfer function are different. The inband region is a standard cosine function whereas the out-of-band region is a hyperbolic cosine function. This requires checking to determine whether the frequency used for calculation is in-band or out-of-band. Note that the "esqr" term is different than that of the Chebyshev Type 1 filter.

 

Public Function Cheby2LPGain(f As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

  fRel = fU / f

  esqr = -10 ^ (Ripple / 10)

  esqr = esqr / (1 - esqr ^ 2)

  If fRel < 1 Then     'In-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else     'Out-of-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby2LPGain = -10 * WorksheetFunction.Log10((1 + esqr * Cn ^ 2) / (esqr * Cn ^ 2))

End Function

 

Public Function Cheby2HPGain(f As Double, fL As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

  fRel = f / fL

  esqr = -10 ^ (Ripple / 10)

  esqr = esqr / (1 - esqr ^ 2)

  If fRel < 1 Then ' Out-of-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else ' Inband

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby2HPGain = -10 * WorksheetFunction.Log10((1 + esqr * Cn ^ 2) / (esqr * Cn ^ 2))

End Function

 

Public Function Cheby2BPGain(f As Double, fL As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

Dim fo As Double

Dim BW As Double

  fo = Sqr(fU * fL)

  BW = (fU - fL) / fo

  fRel = Abs(((f / fo) - (fo / f)) / BW)

  fRel = 1 / fRel

  esqr = -10 ^ (Ripple / 10)

  esqr = esqr / (1 - esqr ^ 2)

  If (f < fL) Or (f > fU) Then     'Out-of-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  Else     'In-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  End If

  Cheby2BPGain = -10 * WorksheetFunction.Log10((1 + esqr * Cn ^ 2) / (esqr * Cn ^ 2))

End Function

 

Public Function Cheby2BSGain(f As Double, fL As Double, fU As Double, N As Double, Ripple As Double) As Double

Dim fRel As Double

Dim esqr As Double

Dim Cn As Double

Dim fo As Double

Dim BW As Double

  fo = Sqr(fU * fL)

  BW = (fU - fL) / fo

  fRel = Abs(((f / fo) - (fo / f)) / BW)

  esqr = -10 ^ (Ripple / 10)

  esqr = esqr / (1 - esqr ^ 2)

  If (f < fL) Or (f > fU) Then     'Out-of-band

    Cn = WorksheetFunction.Cosh(N * WorksheetFunction.Acosh(fRel))

  Else     'In-band

    Cn = Cos(N * WorksheetFunction.Acos(fRel))

  End If

  Cheby2BSGain = -10 * WorksheetFunction.Log10((1 + esqr * Cn ^ 2) / (esqr * Cn ^ 2))

End Function

 

Bessel Filters

Bessel filters exhibit lower out-of-band attenuation rates than Butterworth and Chebyshev, but they have very flat inband linear phase and flat group delay. Textbook suggest a general normalizing factor of √(2N − 1) * ln2) to place the 3 dB corner frequency at the specified cutoff frequency. My experimentation has found that for filter orders 3 through 15, √(1.281.3N - 1) comes closer to 3 dB at the cutoff frequency. Use whatever factor you are most comfortable with.

 

Public Function BesselLPGain(f As Double, fU As Double, N As Double) As Double

Dim Im(0 To 25) As Double

Dim Re(0 To 25) As Double

Dim Bn As Double

Dim Pn As Double

Dim c(0 To 25) As Double

Dim fRel As Double

Dim fAdj As Double

Dim k As Integer

  fAdj = (1.28 ^ 1.3 * N - 1) ^ 0.5

  fRel = f / fU * fAdj

  Bn = 0

  Pn = 0

  For k = 0 To N

  c(k) = WorksheetFunction.Fact(2 * N - k) / _

            (2 ^ (N - k) * WorksheetFunction.Fact(k) * WorksheetFunction.Fact(N - k))

  If WorksheetFunction.IsOdd(k) Then

    If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(Abs(k - 1) / 2, 0)) Then

      Im(k) = -c(k)

    Else

      Im(k) = c(k)

    End If

    Pn = Pn + fRel ^ k * Im(k)

  Else

    If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(k / 2, 0)) Then

      Re(k) = -c(k)

    Else

      Re(k) = c(k)

    End If

    Bn = Bn + fRel ^ k * Re(k)

  End If

  Next k

  BesselLPGain = 20 * WorksheetFunction.Log10(c(0) / (Bn ^ 2 + Pn ^ 2) ^ 0.5)

End Function

 

Public Function BesselHPGain(f As Double, fL As Double, N As Double) As Double

Dim Im(0 To 25) As Double

Dim Re(0 To 25) As Double

Dim Bn As Double

Dim Pn As Double

Dim c(0 To 25) As Double

Dim fRel As Double

Dim fAdj As Double

Dim k As Integer

  fAdj = (1.28 ^ 1.3 * N - 1) ^ 0.5

  fRel = fL / f * fAdj

  Bn = 0

  Pn = 0

  For k = 0 To N

  c(k) = WorksheetFunction.Fact(2 * N - k) / _

            (2 ^ (N - k) * WorksheetFunction.Fact(k) * WorksheetFunction.Fact(N - k))

  If WorksheetFunction.IsOdd(k) Then

    If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(Abs(k - 1) / 2, 0)) Then

      Im(k) = -c(k)

    Else

      Im(k) = c(k)

    End If

    Pn = Pn + fRel ^ k * Im(k)

  Else

    If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(k / 2, 0)) Then

      Re(k) = -c(k)

    Else

      Re(k) = c(k)

    End If

    Bn = Bn + fRel ^ k * Re(k)

  End If

  Next k

  BesselHPGain = 20 * WorksheetFunction.Log10(c(0) / (Bn ^ 2 + Pn ^ 2) ^ 0.5)

End Function

 

Public Function BesselBPGain(f As Double, fL As Double, fU As Double, N As Double) As Double

Dim Im(0 To 25) As Double

Dim Re(0 To 25) As Double

Dim Bn As Double

Dim Pn As Double

Dim c(0 To 25) As Double

Dim fRel As Double

Dim fAdj As Double

Dim k As Integer

Dim fo As Double

Dim BW As Double

  fAdj = (1.28 ^ 1.3 * N - 1) ^ 0.5

  fo = (fU * fL) ^ 0.5

  BW = (fU - fL) / fo

  fRel = (((f / fo) - (fo / f)) / BW) * fAdj

  Bn = 0

  Pn = 0

  For k = 0 To N

    c(k) = WorksheetFunction.Fact(2 * N - k) / _

              (2 ^ (N - k) * WorksheetFunction.Fact(k) * WorksheetFunction.Fact(N - k))

    If WorksheetFunction.IsOdd(k) Then

      If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(Abs(k - 1) / 2, 0)) Then

        Im(k) = -c(k)

      Else

        Im(k) = c(k)

      End If

      Pn = Pn + fRel ^ k * Im(k)

    Else

      If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(k / 2, 0)) Then

        Re(k) = -c(k)

      Else

        Re(k) = c(k)

      End If

    Bn = Bn + fRel ^ k * Re(k)

  End If

  Next k

  BesselBPGain = 20 * WorksheetFunction.Log10(c(0) / (Bn ^ 2 + Pn ^ 2) ^ 0.5)

End Function

 

Public Function BesselBSGain(f As Double, fL As Double, fU As Double, N As Double) As Double

Dim Im(0 To 25) As Double

Dim Re(0 To 25) As Double

Dim Bn As Double

Dim Pn As Double

Dim c(0 To 25) As Double

Dim fRel As Double

Dim fAdj As Double

Dim k As Integer

Dim fo As Double

Dim BW As Double

  fAdj = (1.28 ^ 1.3 * N - 1) ^ 0.5

  fo = (fU * fL) ^ 0.5

  BW = (fU - fL) / fo

  fRel = 1 / (((f / fo) - (fo / f)) / BW) * fAdj

  Bn = 0

  Pn = 0

  For k = 0 To N

    c(k) = WorksheetFunction.Fact(2 * N - k) / _

              (2 ^ (N - k) * WorksheetFunction.Fact(k) * WorksheetFunction.Fact(N - k))

    If WorksheetFunction.IsOdd(k) Then

      If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(Abs(k - 1) / 2, 0)) Then

        Im(k) = -c(k)

      Else

        Im(k) = c(k)

      End If

    Pn = Pn + fRel ^ k * Im(k)

  Else

    If WorksheetFunction.IsOdd(WorksheetFunction.RoundDown(k / 2, 0)) Then

      Re(k) = -c(k)

    Else

      Re(k) = c(k)

    End If

    Bn = Bn + fRel ^ k * Re(k)

  End If

  Next k

  BesselBSGain = 20 * WorksheetFunction.Log10(c(0) / (Bn ^ 2 + Pn ^ 2) ^ 0.5)

End Function

 

 

Posted January 8, 2019

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