`DASMTX` Delay-And-Sum matrix

`DASMTX` returns a Delay-And-Sum matrix for linear beamforming.

## Contents

## Syntax

`M = DASMTX(SIG,X,Z,DELAYS,PARAM)` returns the `numel(X)` -by- `numel(SIG)` delay-and-sum DAS matrix. The matrix `M` can be used to beamform `SIG` (RF or I/Q signals) at the points specified by `X` and `Z`.

`M = DASMTX(SIG,X,Z,PARAM,[X0 Z0])` assumes that the transmit
originates from a virtual source located at `(X0,Z0)`. You may prefer
this syntax after using `TXDELAY(X0,Z0,PARAM)`. This syntax is
recommended for focused transmits.

**TRY IT!** Enter `dasmtx` in the command window for an example.

Because the signals in `SIG` are not required (only its size is needed) to create `M`, the following syntax is recommended:

`M = DASMTX(size(SIG),X,Z,DELAYS,PARAM)`

**!IMPORTANT!** -- With this syntax, use `M = DASMTX(1i*size(SIG),...)` to return a ** complex DAS matrix for I/Q data**.

`DELAYS` are the transmit time delays (in s). The number of elements in `DELAYS` must be the number of elements in the array (which is equal to `size(SIG,2)`). If a sub-aperture was used during transmission, use `DELAYS(i) = NaN` if element # `i` of the linear array was off.

`PARAM` is a structure that contains the parameter values required for DAS beamforming (see below for details).

`DASMTX` versus `DAS`

`DASMTX` is also called by `DAS`. `DASMTX` thus returns the same beamformed results as `DAS`.

**Beamforming using**:`DAS`

`bfSIG = das(SIG,x,z,delays,param)`

**Beamforming using**:`DASMTX`

`M = dasmtx(size(SIG),x,z,delays,param);`

`bfSIG = M*SIG(:);`

`bfSIG = reshape(bfSIG,size(x));`

`M` is a large sparse matrix. Computing `M` can be much more cost-effective than using `DAS` if you need to beamform several `SIG` matrices, because `M` needs to be determined only once.

Let us consider that a series `SIG{1}`, `SIG{2}` ... `SIG{N}` of ultrasound matrices have been generated by sending similar wavefronts with the same ultrasound array. These signals `SIG{i}` are stacked in a 3D array `sig3D` so that `sig3D(:,:,i) = SIG{i}`. To beamform these data with a delay-and-sum approach, the following can be used.

To obtain the DAS matrix, use `dasmtx`:

`M = dastmtx([size(sig3D,1) size(sig3D,2)],x,z,delays,param)` **or**

`M = dastmtx(1i*[size(sig3D,1) size(sig3D,2)],...)` **for I/Q data**.

The beamformed signals are then calculated by a matrix multiplication:

`bfSIG3D = M*reshape(sig3D,[],size(sig3D,3))`

It is necessary to reshape the beamformed matrix array:

`bfSIG3D = reshape(bfSIG3D,size(x,1),size(x,2),[])`

You can also consider saving the DAS matrix `M` in a MAT file and loading it when needed. The previous syntax is generally much faster than:

`for k = 1:N, bfSIG{k} = das(SIG{k},x,z,delays,param); end`

## Other syntaxes

`DASMTX(SIG,X,Z,PARAM)` uses `DELAYS = param.TXdelay`.

`DASMTX(...,METHOD)` specifies the interpolation method. The available methods are decribed in NOTE #3 below.

`[M,PARAM] = DASMTX(...)` also returns the structure `PARAM` with the default values.

## The structure `PARAM`

`PARAM` is a structure that contains the following fields:

`PARAM.fs`: sampling frequency (in Hz,**required**)`PARAM.pitch`: pitch of the transducer (in m,**required**)`PARAM.fc`: center frequency (in Hz,**required**for I/Q signals)`PARAM.radius`: radius of curvature (in m). The default is`Inf`(rectilinear array)`PARAM.TXdelay`: transmission law delays (in s, required if the vector`DELAYS`is not given)`PARAM.c`: longitudinal velocity (in m/s, default = 1540 m/s)`PARAM.t0`: start time for reception (in s, default = 0 s)

**A note on the f-number**

The f-number is defined by the ratio (depth)/(aperture size). A null f-number, i.e. `PARAM.fnumber = 0`, means that the full aperture is used during DAS-beamforming. This might be a suboptimal strategy since the array elements have some directivity.

Use `PARAM.fnumber = []` to obtain an "optimal" f-number, which is estimated from the element directivity (and depends on fc, bandwidth, element width):

`PARAM.fnumber`: reception f-number (default = 0, i.e. full aperture)`PARAM.width`: element width (in m, required if`PARAM.fnumber = []`)**or**`PARAM.kerf`: kerf width (in m, kerf = pitch-width, required if`PARAM.fnumber`= [])`PARAM.bandwidth`: pulse-echo 6dB fractional bandwidth (in %). The default is 60% (used only if`PARAM.fnumber = []`).

## Advanced syntax for vector Doppler

`PARAM.RXangle`: reception angles (in rad, default = 0)

This option can be used for vector Doppler. Beamforming with at least two (sufficiently different) reception angles enables different Doppler directions and, in turn, vector Doppler.

## Notes

**NOTE #1:**`X`- and`Z`-axes

The `X` axis is PARALLEL to the transducer and points from the first (leftmost) element to the last (rightmost) element (`X = 0` at the CENTER of the transducer).

The `Z` axis is PERPENDICULAR to the transducer and points downward (`Z = 0` at the level of the transducer, `Z` increases as depth increases). See the figure below.

For a **convex** array, the `X` axis is parallel to the chord and `Z = 0` at the level of the chord.

**NOTE #2:**

`DASMTX` uses a standard delay-and-sum. It is a linear operator. Phase rotations are included if I/Q (complex) signals are beamformed.

**NOTE #3: interpolation methods**

By default `DASMTX` uses a linear interpolation to generate the DAS matrix. To specify the interpolation method, use `DASMTX(...,METHOD)`, with `METHOD` being:

`'nearest'`: nearest neighbor interpolation`'linear'`: (default) linear interpolation`'quadratic'`: quadratic interpolation`'lanczos3'`: 3-lobe Lanczos (windowed sinc) interpolation`'5points'`: 5-point least-squares parabolic interpolation`'lanczos5'`: 5-lobe Lanczos (windowed sinc) interpolation

The linear interpolation (it is a 2-point method) returns a matrix twice denser than the nearest-neighbor interpolation. It is 3, 4, 5, 6 times denser for `'quadratic'`, `'lanczos3'`, `'5points'`, `'lanczos5'`, respectively (they are 3-to-6-point methods).

## Uniform linear array (ULA)

The ** pitch** is defined as the center-to-center distance between two adjacent elements. The

**width is the distance that separates two adjacent elements. They are constant for a uniform linear array (ULA).**

*kerf*Some functions of the MUST toolbox can also consider curved (convex) ULAs.

## The DAS matrix

The DAS matrix is very sparse. It is complex when beamforming I/Q signals, which should be the preferred approach.

Here is an example of a DAS matrix:

## Example #1: Beamform signals from a phased array

This example shows how to simulate RF signals then beamform I/Q signals

Download the properties of a 2.7-MHz 64-element cardiac phased array in a structure `param` by using `GETPARAM`.

```
param = getparam('P4-2v');
```

Calculate the transmit delays to generate a non-tilted 80-degrees wide circular wave.

width = 60/180*pi; % width angle in rad txdel = txdelay(param,0,width); % in s

Create the scatterers of a 12-cm-by-12-cm phantom.

xs = rand(1,50000)*12e-2-6e-2; zs = rand(1,50000)*12e-2; idx = hypot(xs,zs-.05)<1e-2; xs(idx) = []; % create a 1-cm-radius hole zs(idx) = []; RC = 3+randn(size(xs)); % reflection coefficients

Simulate RF signals by using `SIMUS`.

```
param.fs = 4*param.fc; % sampling frequency
RF = simus(xs,zs,RC,txdel,param);
```

Demodulate the RF signals.

IQ = rf2iq(RF,param);

Create a 256x256 80-degrees wide polar grid with `IMPOLGRID`.

[x,z] = impolgrid([256 256],10e-2,pi/3,param);

Create the DAS matrix.

```
Mdas = dasmtx(1i*size(IQ),x,z,txdel,param);
spy(Mdas(1:10000,1:10000))
title('A close up of the DAS matrix')
```

Beamform the I/Q signals.

IQb = Mdas*IQ(:); IQb = reshape(IQb,size(x));

Create the ultrasound image with `BMODE`.

```
B = bmode(IQb,30); % log-compressed B-mode image with a -30 dB range
```

Display the ultrasound image

pcolor(x*100,z*100,B) c = colorbar; c.YTick = [0 255]; c.YTickLabel = {'-30 dB','0 dB'}; colormap gray title('A simulated ultrasound image') ylabel('[cm]') shading interp axis equal ij tight set(gca,'XColor','none','box','off')

## Example #2: Beamform signals from a convex array

This example shows how to simulate RF signals with a convex array and how to beamform the I/Q signals with `DAS`.

Download the properties of a 2.7-MHz 64-element cardiac phased array in a structure `param` by using `GETPARAM`.

```
param = getparam('C5-2v');
```

Create the scatterers of a 17-cm-by-10-cm cyst phantom.

xs = rand(1,100000)*17e-2-8.5e-2; zs = rand(1,100000)*10e-2; idx = hypot(xs-.02,zs-.04)<1e-2; % create a 1-cm-radius hole xs(idx) = []; zs(idx) = []; RC = 3+randn(size(xs)); % reflection coefficients idx = hypot(xs+.02,zs-.06)<1e-2; % create a 1-cm-radius cyst RC(idx) = RC(idx)+10;

Simulate RF signals by using `SIMUS`.

txdel = zeros(1,128); % transmit delays param.fs = 4*param.fc; % sampling frequency opt.ElementSplitting = 1; % to make simulations faster RF = simus(xs,zs,RC,txdel,param,opt);

Demodulate the RF signals.

IQ = rf2iq(RF,param);

Create a 256x256 polar grid with `IMPOLGRID`.

[x,z] = impolgrid([256 256],10e-2,param);

It is recommended to use an "optimal" f-number.

param.fnumber = [];

Create the DAS matrix with `DASMTX`.

Mdas = dasmtx(1i*size(IQ),x,z,txdel,param);

Beamform the I/Q signals.

IQb = Mdas*IQ(:); IQb = reshape(IQb,size(x));

Create the ultrasound image with `BMODE`.

```
B = bmode(IQb,30); % log-compressed B-mode image with a -30 dB range
```

Display the ultrasound image

pcolor(x*100,z*100,B) c = colorbar; c.YTick = [0 255]; c.YTickLabel = {'-30 dB','0 dB'}; colormap gray title('A simulated ultrasound image') ylabel('[cm]') shading interp axis equal ij tight set(gca,'XColor','none','box','off')

## See also

cite, das, dasmtx3, simus, txdelay

## References

- Perrot V, Polichetti M, Varray F, Garcia D.
**So you think you can DAS? A viewpoint on delay-and-sum beamforming**.*Ultrasonics*, 2021; 111:106309. (**PDF**)

- If you use
`PARAM.RXangle`for vector Doppler: Madiena C, Faurie J, Porée J, Garcia D.**Color and vector flow imaging in parallel ultrasound with sub-Nyquist sampling**.*IEEE TUFFC*, 2018;65:795-802. (**PDF**)

## About the author

Damien Garcia, Eng., Ph.D. INSERM researcher Creatis, University of Lyon, France

**websites**: **BioméCardio**, **MUST**

## Date modified