1. Wavelet Transform
Presented by
Imane Hafnaoui

2. Fourrier Transform Limitations
FT shows what frequencies exist in a
signal.
2 Hz + 10 Hz + 20Hz 3 600
2 500
Magnitude
Magnitude
1 400
Stationary 0 300
-1 200
-2 100
-3 0
0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25
Time Frequency (Hz)

3. • FT is not good with non-stationary signals
2 Hz + 10 Hz + 20Hz 3 600
2 500
Magnitude
Magnitude
1 400
Stationary
0 300
-1 200
-2 100
-3 0
0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25
Time Frequency (Hz)
0.0-0.4: 2 Hz + 1 250
0.8
0.4-0.7: 10 Hz + 0.6 200
0.7-1.0: 20Hz
Magnitude
Magnitude
0.4
0.2 150
0
Non- -0.2
-0.4
100
Stationary -0.6 50
-0.8
-1 0
0 0.5 1 0 5 10 15 20 25
Time Frequency (Hz)

4. • FT only shows how much of each frequency is
present but not at what time it occurs.
Different in Time Domain
1 150 1 150
0.8 0.8
0.6 0.6
0.4 0.4
Magnitude
Magnitude
100 100
Magnitude
Magnitude
0.2 0.2
0 0
-0.2 -0.2
50 50
-0.4 -0.4
-0.6 -0.6
-0.8 -0.8
-1 0 -1 0
0 0.5 1 0 5 10 15 20 25 0 0.5 1 0 5 10 15 20 25
Time Frequency (Hz) Time Frequency (Hz)
Same in Frequency Domain

5. Problem
• Time – Frequency representation is
needed.
Solution
Wavelet Transform

6. Wavelet Transform
1 t
CWTx ,s x ,s xt dt
s s
Translation Scale
Mother Wavelet

7. Applications
• Image Compression
• Signal De-noising
• Edge and Rupture detection