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Digital Beamforming
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Beamforming
• Manipulation of transmit and receive apertures.
• Trade-off performance/cost to achieve:– Steer and focus the transmit beam.– Dynamically steer and focus the receive beam.– Provide accurate delay and apodization.– Provide dynamic receive control.
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Beam Formation as Spatial Filtering
object propagation beam
formation
• Propagation can be viewed as a process of linear filtering (convolution).• Beam formation can be viewed as an inverse filter (or others, such as a matched filter).
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Implementaiton of Beam Formation
• Delay is simply based on geometry.
• Weighting (a.k.a. apodization) strongly depends on the specific approach.
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Beam Formation - Delay
• Delay is based on geometry. For simplicity, a constant sound velocity and straight line propagation are assumed. Multiple reflection is also ignored.
• In diagnostic ultrasound, we are almost always in the near field. Therefore, focusing is necessary.
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Beam Formation - Delay
• Near field / far field crossover occurs when f#=aperture size/wavelength.
• The crossover also corresponds to the point where the phase error across the aperture becomes significant (destructive).
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Beam Formation - Delay
• In practice, ideal delays are quantized, i.e., received signals are temporally sampled.
• The sampling frequency for fine focusing quality needs to be over 32*f0(>> Nyquist).
• Interpolation is essential in a digital system and can be done in RF, IF or BB.
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Beam Formation - Delay
• RF beamformer requires either a clock frequency well over 100MHz, or a large number of real-time computations.
• BB beamformer processes data at a low clock frequency at the price of complex signal processing.
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Beam Formation - Delay
( , , )sin cos
x RRc
xc
xRci
i i 2 2
2
R
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Beam Formation - RF
n t n tx
c t ti( ) ( )cos
1 2
2 2
21 2
11 1
element i ADC interpolation digital delay sum
matio
n
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Beam Formation - RF
• Interpolation by 2:
Z-1
Z-1
MU
X
1/2
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Beam Formation - RF
• General filtering architecture (interpolation by m):
Delay
Filter 1
Filter 2
Filter m-1
MU
X
Fine delay control
FIFO
Coarse delay control
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Beam Formation - BB
BB tA t
e ej f t j fd( )( ) ( )
22 2
element i ADC demod/LPF
time delay/phase rotation
I Q
I
Q
• The coarse time delay is applied at a low clock frequency, the fine phase needs to be rotated accurately (e.g., by CORDIC).
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Beam Formation - Apodization
• Aperture weighting is often simplified as a choice of apodization type (such as uniform, Hamming, Gaussian, ...etc.)
• Apodization is used to control sidelobes, grating lobes and depth of field.
• Apodization generally can use lower number of bits.• Often used on transmit, but not on receive.
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Range Dependence
• Single channel (delay).
• Single channel (apodization).
• Aperture growth (delay and apodization).
1/R
R
R
R R
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Aperture Growth
• Constant f-number for linear and sector formats.
sector linear
• Use angular response for convex formats.
R R
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Aperture Growth
• Use a threshold level (e.g., -6dB) of an individual element’s two-way response to control the aperture growth for convex arrays.
sin
element response
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Aperture Growth
R
r ’
tan
cos cossin sin
1 R rR r
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Aperture Growth
• Use the threshold angle to control lens opening.
• Channels far away from the center channel contribute little to the coherent sum.
• F-number vs. threshold angle.
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Apodization Issues
• Mainlobe vs. sidelobes (contrast vs. detail).
• Sensitivity (particularly for Doppler modes).
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Apodization Issues
• Grating lobes (near field and under-sampled apertures).
• Clinical evaluation of grating lobe levels.
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Apodization Issues
• Near field resolution. Are more channels better ?
• Depth of field : 2* f-number2*using the /8 criterion).
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Apodization Issues
• Large depth of field - better image uniformity for single focus systems.
• Large depth of field - higher frame rate for multiple focus systems.
• Depth of field vs. beam spacing.
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Synthetic Aperture Imaging
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Synthetic Aperture vs. Phased Array
• Phased array has all N2 combinations.
• Synthetic aperture has only N “diagonal” records.
PA SA
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Synthetic Aperture vs. Phased Array
• Conventional phased array: all effective channels are excited to form a transmit beam. All effective channels contribute to receive beam forming.
• Synthetic aperture: a large aperture is synthesized by moving, or multiplexing a small active aperture over a large array.
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Applications in Medical Imaging
• High frequency ultrasound: High frequency (>20MHz) arrays are difficult to construct.
• Some applications:– Ophthalmology.– Dermatology.– Bio-microscopy.
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Applications in Medical Imaging
• Intra-vascular ultrasound: Majority of the imaging device needs to be integrated into a balloon angioplasty device, the number of connection is desired to be at a minimum.
mul
tiple
xor
T/
R
imag
er
catheter
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Applications in Medical Imaging
• Hand-held scanners: multi-element synthetic aperture imaging can be used for optimal tradeoff between cost and image quality.
scanning direction
defocused beamfocused beam
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Applications in Medical Imaging
• Large 1D arrays: For example, a 256 channel 1D array can be driven by a 64 channel system.
• 1.5D and 2D arrays: Improve the image quality without increasing the system channel number.
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Synthetic Aperture vs. Phased Array
• Phased array has all N2 combinations.
• Synthetic aperture has only N “diagonal” records.
PA SA
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Full Data Set
Transmit
Rec
eiv
e
Transmit
Rec
eiv
e
Phased Array Synthetic Aperture
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Synthetic Aperture vs. Phased Array
• Point spread function:
))sin(sin(
))sin(sin()( 0
kd
kNdch
d
weighting
aperture
2d
aperture
weighting
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Synthetic Aperture vs. Phased Array
• Spatial and contrast resolution:
phased array synthetic aperture
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Synthetic Aperture vs. Phased Array
• Signal-to-noise ratio: SNR is determined by the transmitted acoustic power and receive electronic noise. Assuming the same driving voltage, the SNR loss for synthetic aperture is 1/N.
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Synthetic Aperture vs. Phased Array
• Frame rate: Frame rate is determined by the number of channels for synthetic aperture, it is not directly affected by the spatial Nyquist sampling criterion. Thus, there is a potential increase compared to phased array.
ND
c
2rate frame
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Synthetic Aperture vs. Phased Array
• Motion artifacts: For synthetic aperture, a frame cannot be formed until all data are collected. Thus, any motion during data acquisition may produce severe artifacts.
• The motion artifacts may be corrected, but it imposes further constraints on the imaging scheme.
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Synthetic Aperture vs. Phased Array
• Tissue harmonic imaging: Generation of tissue harmonics is determined by its nonlinearity and instantaneous acoustic pressure. Synthetic aperture is not ideal for such applications.
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Synthetic Aperture vs. Phased Array
• Speckle decorrelation: Based on van Cittert Zernike theorem, signals from non-overlapping apertures have no correlation. Therefore, such synthetic apertures cannot be used for correlation based processing such as aberration correction, speckle tracking and Doppler processing.
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Filter Based Synthetic Focusing
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Motivation• Conventional ultrasonic array imaging system
– Fixed transmit and dynamic receive focusing
– Image quality degradation at depths away from the transmit focal zone
• Dynamic transmit focusing– Fully realize the image quality achievable by an array
system
– Not practical for real-time implementation
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DynTx DynRx FixedTx DynRx
beam pattern
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Motivation• Retrospective filtering technique
– Treat dynamic transmit focusing as a deconvolution problem
– Based on fixed transmit and dynamic receive focusing
• Synthetic transmit and receive focusing– Based on fixed transmit and fixed receive focusing
– System complexity is greatly reduced
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Retrospective Filtering
image focused filter inverse image original
)()( 1idealoofidealoof bpsbpbpbps
•Where s: scattering distribution function, bpoof: out of focused pulse-echo beam pattern, bpideal: ideal pulse-echo beam pattern
• all are a function of (R, sinθ)
Transducer A/DBaseband
Demodulation
Beam Buffer
Range-Dependent
Filter Bank
Image Buffer
Signal Processing
Scan Conversion
Display
BeamformerTransducer A/DBaseband
Demodulation
Beam Buffer
Range-Dependent
Filter Bank
Image Buffer
Signal Processing
Scan Conversion
Display
Beamformer
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Inverse filter
• Spatial Fourier transform relationship– Beam pattern aperture function
• The spectrum of the inverse/optimal filter is the ideal pulse-echo effective aperture divided by the out-of-focused pulse-echo aperture function
• Robust deconvolution– No singular point in the passband of spectrum
– SNR is sufficiently high
• The number of taps equals to the number of beams– Not practical
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Optimal filter• Less sensitive to noise than inverse filter• Filter length can be shorter
)1*()*1()1*1( nnnmnm fBy
the mean squared error(MSE)
dfBdfB H
Minimize MSE
dBdBBBf HHopt 11)(
where, b: the out-of-focused beam pattern, d: desired beam pattern
f: filter coefficients
Convolution matrix form
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Pulse-echo effective apertures
0
2 11
2|)(|)( RR
jkx
exCxC
0
5
10
0
DynTx DynRx
0
0.5
1
0
0.5
1DynRx
0.5
1FixedRx
0
5
10
0
5
10
• The pulse-echo beam pattern is the multiplication of the transmit beam and the receive beam
• The pulse-echo effective aperture is the convolution of transmit and receive apertures
For C.W.
R=Ro
R‡Ro
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Experimental Results DynTx DynRx FixedTx DynRx FixedTx FixedRx b filtered d filtered•a •b •c •d •e
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Experimental Results
-0.15 -0.1 -0.05 0 0.05 0.1 0.15-40
-30
-20
-10
0
sinθ
dB
DynTx DynRx DynRx DynRx Filtered FixedRx Filtered
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Experimental Results•DynTx DynRx •FixedTx DynRx •FixedTx FixedRx
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Homework Hint (Due 4/5 noon)Transmit
Rec
eiv
e
Phased Array
Transmit
Rec
eiv
e
Synthetic Aperture
)),,,(,,(),,,(),(1 1
RTRT
N
i
N
jRTPA jiRtjisRjiWRa
T
T
R
R
)),,,(,,(),,,(),(1
TTTT
N
iTTSA iiRtiisRiiWRa
T
T