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<www.excelunusual.com> 1
A Fourier Transform Model in Excel, part #5
<excelunusual.com> by George Lungu
- This fifth part of the tutorial gives plots of the calculated Fourier transform components for a series of input functions using the model created in the previous sections.
- Some of the input functions are created on the spot. Due to the fact that both the functions and the Fourier transforms are sampled in a finite number of points the results are just approximations of the continuous Fourier transform of actual functions in the continuous domain. You will be able to notice windowing errors in the plots.
- The following time signals are processed: AM, FM, rectangular and cardinal sinusoidal.
<www.excelunusual.com> 2
Fourier transform of a MA signal (amplitude modulated):
-These are the Fourier transform components of a sinusoidal signal with an amplitude of 1 and a frequency of 0.7 modulated in amplitude by another sinusoidal signal having an amplitude of 0.5 and a frequency of 0.1. - The Fourier transform was calculated from a frequency of -1.2 to 1.2 in 1000 equally spaced frequency points. - You can see that the spectrum of the carrier is unchanged but the spectrum of the modulating signal is repeated, one part around carrier frequency and the other around the carrier frequency mirror (the negative of the carrier frequency).
Due to the limited duration of the input sample, windowing effects are visible as ringing.
Input
-2
-1
0
1
2
0 10 20 30 40 50 60 70 80 90 100
Time
Amplitude
0
20
40
60
80
100
120
140
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Phase [degrees]
-150
-50
50
150
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Real
-100
-50
0
50
100
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Imaginary
-150
-100
-50
0
50
100
150
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
<www.excelunusual.com> 3
Fourier transform of a FM signal (frequency modulated):
-These are the Fourier transform components of a sinusoidal signal with an amplitude of 1 and a frequency of 0.7 modulated in frequency by another sinusoidal signal having an amplitude of 0.3 and a frequency of 0.01. - The Fourier transform was calculated from a frequency of -1.2 to 1.2 in 1000 equally spaced frequency points. - This time the spectra of the carrier and modulating signals intermix resulting in a much more complex spectrum than in the case of the MA signal.
Input
-1.5
-1
-0.5
0
0.5
1
1.5
0 20 40 60 80 100 120 140 160 180 200
Time
Amplitude
0
5
10
15
20
25
30
35
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Phase [degrees]
-150
-50
50
150
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Real
-40
-20
0
20
40
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
Imaginary
-40
-20
0
20
40
-1.5 -1 -0.5 0 0.5 1 1.5
Frequency
<www.excelunusual.com> 4
Fourier transform of an aperiodic rectangular signal:
-These are the Fourier transform components of a rectangular aperiodic signal with amplitude of 10 and duration of 10. - The Fourier transform was calculated from a frequency of -0.9 to 0.9 in 1000 equally spaced frequency points. - Looking at the real part of the Fourier transform we observe a good similarity with the cardinal sinus function also named a sinc function ( sinc(x) = sin(x)/x ). This is in line with the exact (theoretical) result for the Fourier transform of an aperiodic rectangular function. If this does not ring a bell you could read more about this elsewhere.
Input
-5
0
5
10
15
0 20 40 60 80 100 120 140 160 180 200
Time
Amplitude
0
20
40
60
80
100
120
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Frequency
Phase [degrees]
-150
-50
50
150
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Frequency
Real
-50
0
50
100
150
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Frequency
Imaginary
-100
-50
0
50
100
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Frequency
<www.excelunusual.com> 5
Fourier transform of a MA signal (amplitude modulated):
- Create a “sinc” input:- These are the Fourier transform components of a sinusoidal signal with an amplitude of 1 and a frequency of 0.7 modulated in frequency by another sinusoidal signal having an amplitude of 0.3 and a frequency of 0.01. - The Fourier transform was calculated from a frequency of -1.2 to 1.2 in 1000 equally spaced frequency points. - You can see that the Fourier transform of a rectangular signal is a cardinal sinus signal and vice versa. This can be said about a general signal too (this is not a proof though). The Fourier transform and the reverse Fourier transform have the same formula (up to a constant)
Input
-0.2
0
0.2
0.4
0.6
0 50 100 150 200 250
Time
Amplitude
0
1
2
3
4
5
6
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Frequency
Phase [degrees]
-150
-50
50
150
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Frequency
Real
-10
-5
0
5
10
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Frequency
Imaginary
-10
-5
0
5
10
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Frequency
<www.excelunusual.com> 6
Time scaling: the influence of a non-repetitive pulse duration on its Fourier transform:
-To produce these results we use an impulse of finite duration centered in origin. The time scale will start at a negative time so that the time of zero will be situated in the middle of the time range. We need to do a small change in the spreadsheet: cell A41: “=-2500*B5”- The rectangular signal duration will be placed in cell W36. The signal is centered in origin will be implemented in range W41:W5040. - Cell W41: “=IF(ABS(A41)<=W$36/2,10,0)”then copy W41 down to W5040
- To the right there are the input signal and the real part of the Fourier transform for three different impulse duration cases (4, 16 and 64).- You can see that the wider the signal the narrower the Fourier spectrum. This confirms the theory, a narrower signal will have more power distributed in the higher frequency range than a wider signal.
Real
-20
-10
0
10
20
30
40
50
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Frequency
Input
-0.2
4.8
9.8
14.8
-150 -100 -50 0 50 100 150
Time
Real
-50
0
50
100
150
200
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Frequency
Input
-0.2
4.8
9.8
14.8
-150 -100 -50 0 50 100 150
Time
Real
-200
-100
0
100
200
300
400
500
600
700
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Frequency
Input
-0.2
4.8
9.8
14.8
-150 -100 -50 0 50 100 150
Time
The end.
