Time-Frequency and Time-Scale Analysis of Doppler Ultrasound Signals

1

• 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. 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
  t0 and v0

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

22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
Time-Scale Analysis (Wavelet Transform)

• ?(t) is the analysing wavelet. a (gt0) 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 aa0j and bn.a0j.
• Dyadic wavelet bases are obtained when a02
• 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 vgt0, 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.

32
(No Transcript)
33
(No Transcript)
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.