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Spectral Analysis Using Excel
P. Tran
John Tyler Community College
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Overview
Calculating Fourier transform using Excel.
Spectral analysis of sound from a guitar.
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Applications of Fourier Transform
Signal processing
Quantum mechanics: solving the time-dependent Schrödinger equation
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Evolution of a Wave Packet in a 1D Potential
𝑖ℏ𝜕𝜓
𝜕𝑡= 𝐻𝜓
𝜓 𝑥, 𝑡 = 𝑒−𝑖𝐻ℏ𝑡 𝜓(𝑥, 0)
𝐻 =(ℏ𝑘)2
2𝑚+ 𝑉(𝑥)
𝜓 𝑥, Δ𝑡 ≅ 𝑒−𝑖ℏ𝑉(𝑥)
∆𝑡2 𝑒−
𝑖ℏℏ𝑘 2
2𝑚 ∆𝑡𝑒−𝑖ℏ𝑉(𝑥)
∆𝑡2 𝜓(𝑥, 0)
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Evolution of a Wave Packet in a 1D Potential
𝜓1 𝑥, Δ𝑡 = 𝑒−𝑖ℏ𝑉(𝑥)
∆𝑡2 𝜓(𝑥, 0)
𝜓2 𝑥, Δ𝑡 = 𝑒−𝑖ℏℏ𝑘 2
2𝑚 ∆𝑡𝜓1 𝑥, Δ𝑡
𝜓 𝑥, Δ𝑡 = 𝑒−𝑖ℏ𝑉(𝑥)
∆𝑡2 𝜓2 𝑥, Δ𝑡
𝜓 𝑥, Δ𝑡 ≅ 𝑒−𝑖ℏ𝑉(𝑥)
∆𝑡2 𝑒−
𝑖ℏℏ𝑘 2
2𝑚∆𝑡𝑒−
𝑖ℏ𝑉(𝑥)
∆𝑡2 𝜓(𝑥, 0)
Diagonal in real space
Diagonal in k space
Diagonal in real space
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Why Excel?
Want a computing platform that all students can have access to.
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Nuts and Bolts
𝑓 𝜔𝑖 = න𝑓(𝑡)𝑒−𝑖𝜔𝑖𝑡 𝑑𝑡
Don’t calculate all components at once: too much typing.
Use iterative option of Excel.
Calculate the ith component during the ith iteration. Save for use later.
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Data
Window
Data × window
cos (nti)
sin (nti)
ti
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Be Careful
Excel calculates from left to right and top down for each iteration.
So make sure instructions are in the correct order.
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Not FFT
Can program to do FFT.
But too complex, and the gain in computation time for the data size is not worth it.
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Tuning Fork (384 Hz)
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 200 400 600 800 1000 1200
F(t)
Index
F(t)
F(t)
Window W(t)
F(t)W(t)
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0800
0 100 200 300 400 500 600
F(w
)
Index
F(w)
384.8Hz
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Usual Standing Wave Experiment
Wouldn’t it be more interesting to do an experiment with a musical instrument?
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Standing Wave on a Guitar
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
0 200 400 600 800 1000 1200
F(t)
Index
F(t)
F(t)
Window W(t)
F(t)W(t)
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
0.0450
0.0500
0 100 200 300 400 500 600
F(w
)
Index
F(w)
Fundamental
2nd harmonic
3rd harmonic
4th harmonic
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Frequency vs 1/Length
y = 56x - 4.5R² = 1.0
80
90
100
110
120
130
140
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4
Freq
uen
cy (
Hz)
1/L (m-1)
Fundamental frequency v 1/length
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Frequency vs Wave Speed
y = 0.94x - 17R² = 0.99
0
20
40
60
80
100
120
140
100 110 120 130 140 150
Freq
uen
cy (
Hz)
Speed (m/s)
Frequency vs Wave Speed
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Strength vs Time
0
10
20
30
40
50
60
0 0.1 0.2 0.3 0.4 0.5
Stre
ngt
h
Time (s)
Strength vs time
Fundamental
2nd harmonic
3rd harmonic
4th harmonic
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What Else?
Pluck the string at different positions to see how that affects the composition of the
different harmonics.
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Benefits Over Old Experiment
Students can see how physics works in a real life application.
Teach the principle of superposition of waves.
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Difficulty
We only have one guitar.
Solution: ask students to bring in their own if they have one.
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Thank you