Digital Beamforming
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.
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).
Implementaiton of Beam Formation
• Delay is simply based on geometry.
• Weighting (a.k.a. apodization) strongly
depends on the specific approach.
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.
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).
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.
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.
Beam Formation - Delay
R x i sin  x i cos 
 ( x i , R , )  

c
c
2Rc
2
R

2
Beam Formation - RF
ADC
interpolation
digital delay
x i cos   1 1 
n ( t 1 ) n ( t 2 )  1 
  
2
c   t 1 t 2 
2
2
su m m atio n
element i
Beam Formation - RF
• Interpolation by 2:
Z-1
1/2
MUX
Z-1
Beam Formation - RF
• General filtering architecture (interpolation by m):
Delay
Filter 2
MUX
Filter 1
FIFO
Coarse delay control
Filter m-1
Fine delay control
Beam Formation - BB
element i
ADC
demod/
LPF
I
time delay/
phase rotation
Q
BB ( t ) 
A (t   )
2
I
e
Q
j2f ( t  )  j2f d 
e
• The coarse time delay is applied at a low clock
frequency, the fine phase needs to be rotated accurately
(e.g., by CORDIC).
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.
Range Dependence
• Single channel (delay).
• Single channel
(apodization).
• Aperture growth (delay
and apodization).
1/R
R
R
R
R
Aperture Growth
• Constant f-number for linear and sector
formats.
R
sector
R
linear
• Use angular response for convex formats.
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.
element response
sin
Aperture Growth
r
R

r
’
 R cos  r cos  
r  tan 
 
 R sin   r sin   
1
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.
Apodization Issues
• Mainlobe vs. sidelobes (contrast vs. detail).
• Sensitivity (particularly for Doppler modes).
Apodization Issues
• Grating lobes (near field and undersampled apertures).
• Clinical evaluation of grating lobe levels.
Apodization Issues
• Near field resolution. Are more channels
better ?
• Depth of field : 2* f-number2*l (using the l/8
criterion).
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.
Synthetic Aperture Imaging
Synthetic Aperture vs. Phased Array
PA
SA
• Phased array has all N2 combinations.
• Synthetic aperture has only N “diagonal”
records.
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.
Applications in Medical Imaging
• High frequency ultrasound: High
frequency (>20MHz) arrays are difficult
to construct.
• Some applications:
– Ophthalmology.
– Dermatology.
– Bio-microscopy.
Applications in Medical Imaging
catheter
imager
T/R
multiplexor
• 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.
Applications in Medical Imaging
• Hand-held scanners: multi-element
synthetic aperture imaging can be used
for optimal tradeoff between cost and
image quality.
defocused beam
focused beam
scanning direction
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.
Synthetic Aperture vs. Phased Array
PA
SA
• Phased array has all N2 combinations.
• Synthetic aperture has only N “diagonal”
records.
Full Data Set
Transmit
Receive
Receive
Transmit
Phased Array
Synthetic Aperture
Synthetic Aperture vs. Phased Array
• Point spread function:
sin( kNd sin(  ))
h( )  c0
sin( kd sin(  ))
weighting
weighting
aperture
aperture
d
2d
Synthetic Aperture vs. Phased Array
• Spatial and contrast resolution:
phased array
synthetic aperture
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.
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.
c
frame rate 
2 ND
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.
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.
Synthetic Aperture vs. Phased Array
• Speckle decorrelation: Based on van Cittert
Zernike theorem, signals from nonoverlapping apertures have no correlation.
Therefore, such synthetic apertures cannot
be used for correlation based processing
such as aberration correction, speckle
tracking and Doppler processing.
Filter Based Synthetic Focusing
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
DynTx DynRx
beam pattern
FixedTx DynRx
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
Retrospective Filtering
( s  bpoof )  (bpideal 1 bpoof )  s  bpideal
original image inverse filter
focused image
•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/D
Beamformer
Baseband
Demodulation
Signal Processing
Scan Conversion
Display
Image
Buffer
Range-Dependent
Filter Bank
Beam
Buffer
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-offocused 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
Optimal filter
• Less sensitive to noise than inverse filter
• Filter length can be shorter
Convolution matrix form
y( mn1*1)  B( mn 1*n )  f ( n*1)
the mean squared error(MSE)
  (B f  d  (B f  d 
H
Minimize MSE
f
opt
 (B B  B H d  ( B  )1 d
H
1
where, b: the out-of-focused beam pattern, d: desired beam pattern
f: filter coefficients
Pulse-echo effective apertures
•
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
DynTx DynRx
receive apertures
1
10
2
For C.W.
jkx  1 1 
0.5
 

5
C ( x ) | C ( x ) | e
2  R R0 
0
1
R=Ro
0.5
R‡Ro
0
1
0.5
0
0
DynRx
10
5
0
FixedRx
10
5
0
Experimental Results
DynTx DynRx
•a
FixedTx DynRx
•b
b filtered
•c
FixedTx FixedRx
d filtered
•d
•e
Experimental Results
0
DynTx DynRx
DynRx
DynRx Filtered
FixedRx Filtered
dB
-10
-20
-30
-40
-0.15
-0.1
-0.05
0
sinθ
0.05
0.1
0.15
Experimental Results
•DynTx DynRx
•FixedTx DynRx
•FixedTx FixedRx
Homework Hint (Due 4/5 noon)
Transmit
Receive
Receive
Transmit
Synthetic Aperture
Phased Array
NT
NR
aPA ( R, )   W (iT , jR , R, )s(iT , jR , t   ( R, , iT , jR ))
iT 1 j R 1
NT
aSA ( R, )  W (iT , iT , R, )s(iT , iT , t   ( R, , iT , iT ))
iT 1