Introduction to Wavelets

David HerrinUniversity of Kentucky

Introduction to Wavelets

Vibro-Acoustics Consortium

July 29, 2021

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July 29, 2021

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What is a Wavelet?

A waveform of limited duration with an average value of zero (i.e. a small wave).

Haar Wavelet Morlet Wavelet Daubechies 2 (db2)

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What is a Wavelet?

Stretching the Haar Wavelet

+1 -1+1 +1 -1 -1+1 +1 +1 +1 -1 -1 -1 -1+1 +1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 -1

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What is a Wavelet?

Time

Frequency

Precision

Time

Frequency

Precision

Heisenberg Uncertainty Principle – It is impossible to know exact frequency content at an exact time.

Same Area

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Wavelet Transform

Time

Frequency

Precision

Intuitively

Time

Scale

Precision

Wavelet

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Music Analogy (i.e. Scale)

Low notes (low frequency) need longer to be correctly generated while high notes (high frequency) can be generated quickly.

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Wavelet Transforms

• Continuous Wavelet Transform • Discrete Wavelet Transform

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Continuous Wavelet Transform Step 1

Take a wavelet and compare it to a section at the start of the original signal.

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Calculate a number 𝐶 that indicates how closely correlated the wavelet is with this section of the signal. The higher 𝐶 is the more similarity. Correlation depends partly on the wavelet selected.

Continuous Wavelet Transform Step 2

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Shift wavelet slightly to the right and repeat steps 1 and 2.

Continuous Wavelet Transform Step 3

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Scale (stretch) the wavelet slightly and repeat again.

Continuous Wavelet Transform Step 4

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Repeat the process for all scales. If we do this at 10 scales and for 400 time steps, we will have a 400 by 10 set of correlation coefficients. A wavelet that resembles what you are looking for will give a strong correlation at some stretch and shift.

Continuous Wavelet Transform Step 5

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Continuous Wavelet Transform

Wavelet Signal

Translation (Position) Parameter

Scale (Dilation) Parameter

Time SignalCorrelation Coeffient

𝐶 𝑎, 𝑏1𝑎

𝑓 𝑡 𝜓𝑡 𝑏𝑎 𝑑𝑡

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Example Non-Stationary Signal

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Example Non-Stationary Signal

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Example Cab Reverberation

Balloon located close to door towards front of the passenger compartment.

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107 Hz

196 Hz

Example Cab Reverberation

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Continuous Wavelet Transform

Smooth shifting and scaling

Use any wavelet you want

Inverse CWT is difficult

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Discrete Wavelet Transform

Signals can be reconstructed after being DWT

Scales are powers of 2

Wavelets must be constructed from digital filters

Daubechies 2 (db2)

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Wavelet FiltersLow Pass Filter Low Pass Reconstruction Filter

High Pass Filter High Pass Reconstruction Filter

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Wavelet Filters

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low-pass

high-pass

Approximation 1000 Samples

Approximation and Details

1000 Samples

Details 1000 Samples

S

D

A

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Multiple Level Decomposition

S

cA1 cD1

cA2 cD2

cA3 cD3

Frequency

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Downsampling and Upsampling

0.12 0.15 0.18 0.15 0.12 0.09 0.06 0.03 0.00 0.03 0.05 0.09 0.12

Original Signal

Downsampled Signal (remove every other data point)

0.12 0.18 0.12 0.06 0.00 0.05 0.12

Upsampled Signal (insert 0’s between every other data point)

0.12 0.00 0.18 0.00 0.12 0.00 0.06 0.00 0.00 0.00 0.05 0.00 0.12

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low-pass

high-pass

Approximation 500 Samples

With Wavelet Filters

1000 Samples

Details 500 Samples

S

cD

cA

Downsample

Downsample

Downsample – Throw away every other term in the data

A

D

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Discrete Wavelet Transform

S

𝐿

𝐻

𝐿

𝐻

S

cA

cD

cA

cD

A

D

A

D

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Wavelet Transforms

• Continuous Wavelet Transform • Discrete Wavelet Transform

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D. L. Fugal, Conceptual Wavelets in Digital Signal Processing: an In-Depth, Practical Approach for the Non-Mathematician, Space and Signals Technical Publishing, San Diego (2009).

Wavelets Reference