Time and Frequency Representations Accompanying presentation Kenan Gençol presented in the course...

Time and Frequency Time and Frequency RepresentationsRepresentations

Accompanying presentationAccompanying presentation

Kenan GençolKenan Gençol

presented in the coursepresented in the courseSignal TransformationsSignal Transformations

instructed byinstructed byProf.Dr. Ömer Nezih GerekProf.Dr. Ömer Nezih Gerek

Department of Electrical and Electronics Engineering, Department of Electrical and Electronics Engineering, Anadolu UniversityAnadolu University

Stationary and nonstationary Stationary and nonstationary signalssignals

A stationary signal

A nonstationary signal

time varying spectral components

spectral components do not change in time

Stationary and nonstationary Stationary and nonstationary signalssignals

Stationary signals consist of spectral components that do Stationary signals consist of spectral components that do not change in time not change in time all spectral components exist at all timesall spectral components exist at all times FT works well for stationary signalsFT works well for stationary signals

However, non-stationary signals consists of time varying However, non-stationary signals consists of time varying spectral componentsspectral components How do we find out which spectral component How do we find out which spectral component appears whenappears when?? FT only provides FT only provides what spectral components exist what spectral components exist , not where in time , not where in time

they are located.they are located. Need some other ways to determine Need some other ways to determine time localization of spectral time localization of spectral

componentscomponents FT identifies all spectral components present in the signal, FT identifies all spectral components present in the signal,

however it does not provide any information regarding the however it does not provide any information regarding the temporal (time)temporal (time) localization of these components.localization of these components.

STFTSTFT

STFT Sliding WindowSTFT Sliding Window

The Wavelet TransformThe Wavelet Transform

An Example: STFT - An Example: STFT - SpectrogramSpectrogram

STFT amplitude spectrum (Spectrogram) of a musical performance

Magnitude (dB)

STFT and Wavelet Spectrogram STFT and Wavelet Spectrogram Comparison – An exampleComparison – An example

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


Time is well-localized but the two lower frequency Time is well-localized but the two lower frequency tones 10 and 13 Hz are not resolved.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


The two low frequencies are now resolved but The two low frequencies are now resolved but now the interruption in the higher- frequency now the interruption in the higher- frequency term 80 Hz is not resolved.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


Both time and frequency are well-localized. Note Both time and frequency are well-localized. Note vertical bars on the ends of the notes reflect the vertical bars on the ends of the notes reflect the sharp cut-off and cut-on of the tones (higher sharp cut-off and cut-on of the tones (higher

frequency content)frequency content)

Continuous wavelet transform 10 Hz

13 Hz

80 Hz interrupted

Vertical bars

STFT and Wavelet Resolution - STFT and Wavelet Resolution - ComparisonComparison

Time

Frequenc

y

DWT (Discrete Wavelet DWT (Discrete Wavelet Transform), a dyadic Transform), a dyadic

decompositiondecomposition Calculating wavelet coefficients at every Calculating wavelet coefficients at every

possible scale is a possible scale is a huge huge amount of workamount of work.. For each of the m scales, CWT perform a For each of the m scales, CWT perform a

convolution on the raw signal of length n.convolution on the raw signal of length n. TheThe CWTCWT return m · n coefficientsreturn m · n coefficients inin time time

O (m · n log(n)).O (m · n log(n)). There is a huge amount of redundancy There is a huge amount of redundancy

and for higher scales, we could use a and for higher scales, we could use a smaller sampling rate.smaller sampling rate.


decompositiondecomposition IIf we choose scales and positions based on f we choose scales and positions based on

powers ofpowers of two two --- so-called - so-called dyadicdyadic scales and scales and positions positions --- then our analysis will be much more - then our analysis will be much more efficient and just as accurate.efficient and just as accurate.

An efficient way to implement this scheme using An efficient way to implement this scheme using filtersfilters..

Instead of stretching the wavelet to get to a Instead of stretching the wavelet to get to a bigger scale, we will compress the original signal.bigger scale, we will compress the original signal.

For that, we need a second wavelet, called the For that, we need a second wavelet, called the scaling functionscaling function. This function is a . This function is a lowpass filterlowpass filter. . The wavelet is complementary filter, The wavelet is complementary filter, a highpass a highpass filterfilter..


decompositiondecompositionScaling function

Wavelet function

Scaling and wavelet functions and their frequency responses


decompositiondecomposition To perform DWT, we start from the signal To perform DWT, we start from the signal

and split the signal in two parts.and split the signal in two parts. DetailsDetails, using the , using the waveletwavelet.. ApproximationApproximation, using the , using the scaling functionscaling function.. We then start back the decomposition from We then start back the decomposition from

the approximated signal.the approximated signal. And again...And again... All the details is our wavelet All the details is our wavelet

transformtransform.. We need to We need to keep the last approximationkeep the last approximation for for

the inverse transform.the inverse transform.


decompositiondecomposition

Decomposition Process


decompositiondecomposition To perform the inverse DWT, we start from To perform the inverse DWT, we start from

the details and last approximation.the details and last approximation. We combine the last approximation with We combine the last approximation with

the last details, and find the seond last the last details, and find the seond last approximation.approximation.

And repeat...And repeat... Both inverse and forward take O(n), faster Both inverse and forward take O(n), faster

than fourier transform.than fourier transform. But DWT restricts us to an octave But DWT restricts us to an octave

frequency resolution.frequency resolution.


decompositiondecomposition

Reconstruction Process

Time and Frequency Representations Accompanying presentation Kenan Gençol presented in the course...

Documents

Transcript of Time and Frequency Representations Accompanying presentation Kenan Gençol presented in the course...