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.


• 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).


• 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.


• 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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x RRc

xc

xRci

i i 2 2

2

R

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

Beam Formation - RF

• Interpolation by 2:

Z-1

Z-1

MU

X

1/2

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

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).

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.

sector linear

• Use angular response for convex formats.

R R

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

Aperture Growth

R

r ’

tan

cos cossin sin

1 R rR r

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 under-sampled apertures).

• Clinical evaluation of grating lobe levels.

Apodization Issues

• Near field resolution. Are more channels better ?

• Depth of field : 2* f-number2*using the /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

• Phased array has all N2 combinations.

• Synthetic aperture has only N “diagonal” records.

PA SA


• 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.


• 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


• Hand-held scanners: multi-element synthetic aperture imaging can be used for optimal tradeoff between cost and image quality.

scanning direction

defocused beamfocused beam


• 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.


• Phased array has all N2 combinations.

• Synthetic aperture has only N “diagonal” records.

PA SA

Full Data Set

Transmit

Rec

eiv

e

Transmit

Rec

eiv

e

Phased Array Synthetic Aperture


• Point spread function:

))sin(sin(

))sin(sin()( 0

kd

kNdch

d

weighting

aperture

2d

aperture

weighting


• Spatial and contrast resolution:

phased array synthetic aperture


• 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.


• 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


• 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.


• Tissue harmonic imaging: Generation of tissue harmonics is determined by its nonlinearity and instantaneous acoustic pressure. Synthetic aperture is not ideal for such applications.


• 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.

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 FixedTx DynRx

beam pattern

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

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

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

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

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

Experimental Results DynTx DynRx FixedTx DynRx FixedTx FixedRx b filtered d filtered•a •b •c •d •e

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

Experimental Results•DynTx DynRx •FixedTx DynRx •FixedTx FixedRx

Homework Hint (Due 4/5 noon)Transmit

Rec

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Phased Array

Transmit

Rec

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e

Synthetic Aperture

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Digital Beamforming

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