1. Time-Frequency and Time-Scale Analysis of Doppler Ultrasound Signals
Lecture 7:
Time-Frequency and Time-Scale Analysis of Doppler Ultrasound Signals

2. Stationar/nonstationary signals
Most biological signals are highly non-stationary and sometimes last only for a short time. Signal analysis methods which assume that the signal is stationary are not appropriate. Therefore time-frequency analysis of such signals is necessary.

3. Some nonstationary signals
Some nonstationary signals. Forward (red) and reverse (blue) flow components are shown (after Hilbert transform process)

4. Typical audio Doppler signals

5. Time-Frequency Analysis...
The time representation is usually the first description of a signal s(t) obtained by a receiver recording variations with time. The frequency representation, which is obtained by the well known Fourier transform (FT), highlights the existence of periodicity, and is also a useful way to describe a signal.

6. Time-Frequency Analysis...
The relationship between frequency and time representations of a signal can be defined as
no frequency information no time information

7. Time-Frequency Analysis
A joint time-frequency representation is necessary to observe evolution of the signal both in time and frequency.
Linear methods (Windowed Fourier Transform (WFT), and Wavelet Transform (WT))
Decomposes a signal into time-frequency atoms.
Computationally efficient
Time-frequency resolution trade-off
Bilinear (quadratic) methods (Wigner-Ville distribution)
Based upon estimating an instantaneous energy distribution using a bilinear operation on the signal.
Computationally intense
Arbitrarily high resolution in time-and frequncy
Cross term interference

8. Windowed Fourier Transform...
g(t): short time analysis window localised around t=0 and v=0

9. Windowed Fourier Transform...
Segmenting a long signal into smaller sections with 128 point and 512 point Hanning window (criticaly sampled).

10. Windowed Fourier Transform...
Assumes the signal is stationary within the analysis window.
Time-frequency tiling is fixed
The WFT is similar to a bank of band-pass filters with constant bandwidth.
Three important WFT parameters for analysis of a particular signal need to be determined: Window type, window size, required overlap ratio

11. Window type...
The FT makes an implicit assumption that the signal within the measured time is repetitive. Most real signals will have discontinuities at the ends of the measured time, and when the FFT assumes the signal repeats it will also assume discontinuities that are not there. Discontinuities will be eliminated by multiplying the signal with a window function.

12. Window type
Some window types and corresponding power spectra

13. Window size...
In a FFT process, there is a well known trade-off between frequency resolution (v) and time resolution (t), which can be expressed as
where W is window length and vs is sampling frequency.
To use the WFT one has to make a trade-off between time-resolution and frequency resolution.

14. Window size
If no overlap is employed, processing NS length data by using W length analysis window will result in a time-frequency distribution having a dimension that almost equals to the dimension of the original signal space (critically sampled WFT). The actual dimension of the time-frequency distribution is
The best combination of t and f depends on the signal being processed and best time-frequency resolution trade-off needs to be determined empirically.

15. Window overlap ratio...
A short duration signal may be lost when a windowing function is used prior to the FFT.
In this case an overlap ratio to some degree must be employed.
In overlapped WFT, the data frames of length W are processed sequentially by sliding the window ‘W-OL’ times at each processing stage, where OL is the number of overlapped samples. Consequently, overlapping FFT windows produces higher dimensional WFTs. In an overlapped WFT process, the dimension of the resultant time-frequency distribution is

16. Window overlap ratio...
The overlapping process introduces a predictable time shift on the actual location of a transient event on the time-frequency plane of the FFT.
The duration of the time shift depends on the overlap ratio used, while the direction of the time shift is dictated by the way that the data are arranged prior to the FFT.
Duration of the time shift can be estimated as ‘(number of overlapped samples/2)sampling time’.
The time shift can be adjusted by adding zeros equally at both ends of the original data array. In this case the dimension of the overlapped WFT is

17. Window overlap ratio
Segmenting a long signal into smaller sections with 512 point Hanning window (different overlap strategies).

18. TFDs with 16, 32, 64 ,128, 256, 256, 512 points windowing
Normalised IP and energy with 16(black), 32(red), 64(green), 128(blue), 256(magenta), 512(cyan) windowing)

19. Normalised IP and energy with 16(black), 32(red), 64(green), 128(blue), 256(magenta), 512(cyan) hanning windowing
TFDs with 16, 32, 64 ,128, 256, 256, 512 points hanning windowing

20. An example of possible embolic signal at the edges of two consecutive frames and related 3d spectrum without a window and with a Hannig window function

21. Linear and Logarithmic sonogram displays of a Doppler signal with possible embolic signal using different overlap ratios

27. Time-Scale Analysis (Wavelet Transform)
*(t) is the analysing wavelet. a (>0) controls the scale of the wavelet. b is the translation and controls the position of the wavelet.
Can be computed directly by convolving the signal with a scaled and dilated version of the wavelet (Frequency domain implementation may increase computational efficiency).
Discrete WT is a special case of the continuous WT when a=a0j and b=n.a0j.
Dyadic wavelet bases are obtained when a0=2
Wavelets are ideally suited for the analysis of sudden short duration signal changes (non-stationary signals).
Decomposes a time series into time-scale space, creating a three dimensional representation (time, wavelet scale and amplitude of the WT coefficients).
Time-frequency (scale) tiling is logarithmic.

28. The wavelet transform properties
It is a linear transformation,
It is covariant under translations:
It is covariant under dilations:
If W(a,b) is the WT of a signal s(t), then s(t) can be restored using the formula:
providing that the Fourier transform of wavelet (t), denoted (v) satisfies the following admissibility condition: ,
which shows that (t), has to oscillate and decay.

29. Morlet Wavelet
One of the original wavelet functions is the Morlet wavelet, which is a locally periodic wavetrain. It is obtained by taking a complex sine wave, and localising it with a Gaussian envelope.
Real (blue) and imaginary (red) componets of Morlet wavelet.

30. Morlet wavelet
where v0 is nondimensional frequency and ususally assumed to be 5 to 6 to satisfy the admissibility condition. Fourier transform and corresponding Fourier wavelength of Morlet wavelet are
H(v) =1 if v>0, H(v)=0 otherwise

31. The basic difference between the WT and the WFT is that when the scale factor a is changed, the duration and the bandwidth of the wavelet are both changed but its shape remains the same.
The WT uses short windows at high frequencies and long windows at low frequencies in contrast to the WFT, which uses a single analysis window. This partially overcomes the time resolution limitation of the WFT.

34. Long (a) and short (b) duration embolic signals and corresponding 2d and 3d plots of the FFT and the CWT results. The FFT window size was 128 point (17.9 ms) with Hanning window and the 64 scales Morlet wavelet was used for the CWT.
A low intensity embolic signal and corresponding 2d and 3d plots of the WFT and the CWT results (- indicates reverse flow direction) for (a) the 128 point FFT and 64 scales CWT, (b) the 32 point FFT and 32 scale CWT.