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CREATING WAVEFORMS WITH THE FOURIER SERIES

A SUM OF SINES AND COSINES

I've always been intrigued with the Fourier Series theory. It says you can generate strange and wonderful waveforms (square, triangle, etc.) simply by summing sine and cosine waveforms together! Mathematically, that looks like this

y(t) = a0  +  a1·cos(2·π·fo·t) + a2·cos(2·π·2·fo·t) + a3·cos(2·π·3·fo·t) + ... +
b1·sin(2·π·fo·t) + b2·sin(2·π·2·fo·t) + b3·sin(2·π·3·fo·t) + ...

where
a0 - the DC offset.
a1 - scales how much of the fundamental frequency fo of a cosine is in the waveform
.
a2 - scales how much of the second harmonic (2*fo) of the cosine is added to the waveform|
a3 - scales the thirds harmonic (3*fo).
b1, b2, b3 - scales the sine wave components added to the waveform.

In a compact form, the series looks like

y(t) = a0  +  Σ  an·cos(2·π·fo·n t) + Σ  bn·sin(2·π·fo·n t)

where Σ simply means sum all the terms for n=1 to ∞.

Let's write a simple VBA function to help us play with waveforms via the Fourier Series. You can download the file . (Also see VBA Basics and other  VBA Topics).

In the file, you first enter the fundamental frequency (fo) and number of Fourier terms (n) you want to use in the waveform creation. You also need a time increment (dT) so you can create a time variable. Lastly, the actual Fourier coefficients are placed in a 2D table. The term a0 represents the DC offset (bn is not used.)

 dT 0.0001 Time increment (Sampling period) fo 100 fundamental freq ntot 10 Use n terms in the waveform calc. Terms an bn 0 0.00 na DC Term (a0) 1 0.00 1.27 fund freq term 2 0.00 0.00 2nd harmonic 3 0.00 0.42 3rd 4 0.00 0.00 ,,, 5 0.00 0.25 6 0.00 0.00 7 0.00 0.18 8 0.00 0.00 9 0.00 0.14 10 0.00 0.00

The time column is generated by adding dT to the previous sample.

 t Vo 0.0000 0.00 0.0001 0.39 0.0002 0.73 0.0003 0.98 0.0004 1.13 0.0005 1.17

The actual waveform Vo gets created by the function = FourSeries( t, ntot , a0:b10, fo ). Using actual cell references, it looks like = FourSeries(A31,\$B\$13,\$B\$16:\$C\$26,\$B\$12). Notice you pass the entire range of cells holding coefficients a0 through b10 to the function using \$B\$16:\$C\$26. However, you specify how many terms to actually use via the parameter ntot.

THE FOURIER SERIES FUNCTION

Here's the VBA code that creates the waveform based on the Fourier coefficients. To see the VBA code, hit ALT-F11 and double click on the Modules > Module1 in the VBA Project window. This opens the code window for this module.

 Function FourSeries(t, ntot, FCoeff, fo) ' Calculate waveform based on Fourier Coefficients listed in table ' Coefficients table: 1st col is an, 2nd col is bn ' ' t - time ' N - number of Fourier coefficients to include in calculation. ' FCoeff - 2D cell range of table with coeff ' fo - fundamental frequency Dim y, a0, an, bn As Double Dim R, C, n As Integer R = FCoeff.Row ' get start position of coeff table C = FCoeff.Column a0 = Cells(R, C) ' get DC offset ' Sum all sine and cosine terms at time t y = a0 ' initialize with a0. For n = 1 To ntot    ' Get Fourier Coeff for frequency n*fo    an = Cells(R + i, C)    bn = Cells(R + i, C + 1)    ' Sum the sine and cosine using Fourier coeff as scale factors    y = y + an*Cos(2*3.1415*(fo*n)*t) + bn*Sin(2*3.1415*(fo*n)*t) Next n ' assign y to return variable of function FourSeries = y End Function

First, we'll get the row and column location of the coefficient table. The range of cells for the table is passed to the variable FCoeff. VBA has two handy properties ( *.Row and *.Col ) that returns the Row and Column of a single cell, or the first cell in a range of cells.

R = FCoeff.Row ' get start position of coeff table
C = FCoeff.Column

Now, have the values R = 16 and C = 2. This allows us to get the a0 coefficient using the Cells( ) property.

a0 = Cells(R, C) ' get DC offset

We'll initialize the waveform (variable y) with the DC offset. Finally, we'll create a loop that sums together the sine and cosine waves at the fundamental frequency fo and higher harmonics fo*n up to n=ntot. Notice, we get the an, bn coefficients for each value of n using the Cells( ) property along with R and C.

For n = 1 To ntot

' Get Fourier Coeff for frequency n*fo
an = Cells(R + n, C)
bn = Cells(R + n, C + 1)

' Sum the sine and cosine using Fourier coeff as scale factors
y = y + an*Cos(2*3.1415*(fo*n)*t) + bn*Sin(2*3.1415*(fo*n)*t)

Next n

Finally, we'll pass the waveform value y calculated at time t to the name FourSeries to be returned by the function.

WAVEFORMS

Okay, let's start creating some waveforms. To generate a square wave, enter the bn coefficients below.

 Terms an bn 0 0.00 na 1 0.00 1.27 2 0.00 0.00 3 0.00 0.42 4 0.00 0.00 5 0.00 0.25 6 0.00 0.00 7 0.00 0.18 8 0.00 0.00 9 0.00 0.14 10 0.00 0.00

As you enter each coefficient, you can see the waveform start approaching the shape of a square wave.

To create a sawtooth waveform, simply go to the bn column and enter 1/n for each of the terms - 1, 1/2, 1/3 and so on.

Here are some other sine type waveforms to try.

Sine signal                      a1= 0,    b1= 1

Inverted sine                  a1= 0,    b1= -1

Cosine                            a1= 1,    b1= 0

Cosine w/ -45 deg shift  a1= 0.7  b1= 0.7

Also try varying some other parameters.

Change fundamental frequency fo to 200.

Change sampling period to 0.0002 or 0.00005.

Change DC Offset (a0) to 5.

Add random values to harmonic terms 0 through 10.

In another topic we'll learn how to perform Fourier Analysis. This analysis does the opposite of the Fourier Series. It takes an arbitrary waveform and then extracts the Fourier Coefficients.