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
Fourier_Series.xls. (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 ALTF11 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.
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