1. Time and Frequency Representations Accompanying presentation Kenan Gençol presented in the course Signal Transformations instructed by Prof.Dr. Ömer Nezih Gerek Department of Electrical and Electronics Engineering, Anadolu University

2. Stationary and nonstationary signals A stationary signal A nonstationary signal time varying spectral components spectral components do not change in time

3. Stationary and nonstationary signals Stationary signals consist of spectral components that do not change in time Stationary signals consist of spectral components that do not change in time all spectral components exist at all times all spectral components exist at all times FT works well for stationary signals FT works well for stationary signals However, non-stationary signals consists of time varying spectral components However, non-stationary signals consists of time varying spectral components How do we find out which spectral component appears when? How do we find out which spectral component appears when? FT only provides what spectral components exist, not where in time they are located. FT only provides what spectral components exist, not where in time they are located. Need some other ways to determine time localization of spectral components Need some other ways to determine time localization of spectral components FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components. FT identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of these components.

4. STFT

5. STFT Sliding Window

6. The Wavelet Transform

8. An Example: STFT - Spectrogram STFT amplitude spectrum (Spectrogram) of a musical performance Magnitude (dB)

9. STFT and Wavelet Spectrogram Comparison – An example This section gives a comparison of STFT and wavelet spectrograms of an artificial sinusoidal signal consisting of an interrupted 80Hz pure tone superimposed over pure tones of 10 and 13Hz as an example. This section gives a comparison of STFT and wavelet spectrograms of an artificial sinusoidal signal consisting of an interrupted 80Hz pure tone superimposed over pure tones of 10 and 13Hz as an example.

10. STFT and Wavelet Spectrogram Comparison – An example Time is well-localized but the two lower frequency tones 10 and 13 Hz are not resolved. Time is well-localized but the two lower frequency tones 10 and 13 Hz are not resolved. Short-time Fourier (Gabor) transform with a narrow window h=0.05 s. 80 Hz interrupted 10 and 13 Hz are not resolved

11. STFT and Wavelet Spectrogram Comparison – An example The two low frequencies are now resolved but now the interruption in the higher- frequency term 80 Hz is not resolved. The two low frequencies are now resolved but now the interruption in the higher- frequency term 80 Hz is not resolved. Short-time Fourier (Gabor) transform with a wide window h=0.3 s. 10 Hz 13 Hz Interruption is not resolved

12. STFT and Wavelet Spectrogram Comparison – An example Both time and frequency are well-localized. Note vertical bars on the ends of the notes reflect the sharp cut-off and cut-on of the tones (higher frequency content) Both time and frequency are well-localized. Note vertical bars on the ends of the notes reflect the sharp cut-off and cut-on of the tones (higher frequency content) Continuous wavelet transform 10 Hz 13 Hz 80 Hz interrupted Vertical bars

13. STFT and Wavelet Resolution - Comparison Time Frequency

14. DWT (Discrete Wavelet Transform), a dyadic decomposition Calculating wavelet coefficients at every possible scale is a huge amount of work. Calculating wavelet coefficients at every possible scale is a huge amount of work. For each of the m scales, CWT perform a convolution on the raw signal of length n. For each of the m scales, CWT perform a convolution on the raw signal of length n. The CWT return m · n coe ffi cients in time O (m · n log(n)). The CWT return m · n coe ffi cients in time O (m · n log(n)). There is a huge amount of redundancy and for higher scales, we could use a smaller sampling rate. There is a huge amount of redundancy and for higher scales, we could use a smaller sampling rate.

15. DWT (Discrete Wavelet Transform), a dyadic decomposition If we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. If we choose scales and positions based on powers of two -- so-called dyadic scales and positions -- then our analysis will be much more efficient and just as accurate. An efficient way to implement this scheme using filters. An efficient way to implement this scheme using filters. Instead of stretching the wavelet to get to a bigger scale, we will compress the original signal. Instead of stretching the wavelet to get to a bigger scale, we will compress the original signal. For that, we need a second wavelet, called the scaling function. This function is a lowpass filter. The wavelet is complementary filter, a highpass filter. For that, we need a second wavelet, called the scaling function. This function is a lowpass filter. The wavelet is complementary filter, a highpass filter.

16. DWT (Discrete Wavelet Transform), a dyadic decomposition Scaling function Wavelet function Scaling and wavelet functions and their frequency responses

17. DWT (Discrete Wavelet Transform), a dyadic decomposition To perform DWT, we start from the signal and split the signal in two parts. To perform DWT, we start from the signal and split the signal in two parts. Details, using the wavelet. Details, using the wavelet. Approximation, using the scaling function. Approximation, using the scaling function. We then start back the decomposition from the approximated signal. We then start back the decomposition from the approximated signal. And again... And again... All the details is our wavelet transform. All the details is our wavelet transform. We need to keep the last approximation for the inverse transform. We need to keep the last approximation for the inverse transform.

18. DWT (Discrete Wavelet Transform), a dyadic decomposition Decomposition Process

19. DWT (Discrete Wavelet Transform), a dyadic decomposition To perform the inverse DWT, we start from the details and last approximation. To perform the inverse DWT, we start from the details and last approximation. We combine the last approximation with the last details, and find the seond last approximation. We combine the last approximation with the last details, and find the seond last approximation. And repeat... And repeat... Both inverse and forward take O(n), faster than fourier transform. Both inverse and forward take O(n), faster than fourier transform. But DWT restricts us to an octave frequency resolution. But DWT restricts us to an octave frequency resolution.

20. DWT (Discrete Wavelet Transform), a dyadic decomposition Reconstruction Process