Page 278 - Biomedical Engineering and Design Handbook Volume 2, Applications

256 DIAGNOSTIC EQUIPMENT DESIGN

Assuming that at location p there is a point target with reflectivity W , then the signal after the
p
summer in Fig. 9.3 can be described by
N An ( ) rn ( )
⎡
Rt ( ) = W p∑ R Tt − t R ( n) + ⎤ (9.2)
rn ( ) ⎣ ⎢ c ⎦ ⎥
n=1
where A (n) = the pressure amplitude contribution of the nth element to echoes from point p
T
T(t) = given by Eq. (9.1)
t (n) = the receive focusing delay for element n
R
The remaining parameters of Eq. (9.2) were defined in Eq. (9.1). It should be noted that the A (n)
T
and A (n) terms in Eqs. (9.1) and (9.2) will, in general, be different since the transmit and receive
R
operation need not be symmetric. The terms include tissue attenuation, element sensitivity variation,
and transmit or receive apodizations.
It might be useful at this point to discuss several methods by which the receive delays for either
focusing or beam steering are implemented. The previous paragraph refers to the use of delay lines
for this purpose. Analog delay lines are an older, albeit a very cost-effective method. However,
lumped-constant delay lines do suffer from several limitations. Among these is the limited bandwidth
associated with longer delay lines. Delays needed for focusing for most apertures are less than 0.5
μs; however, for phased-array beam steering (see below) they may be as long as 8 μs for larger aper-
tures required for 2.5- to 3.5-MHz operation or up to 5 μs required for 5- to 7-MHz operation. Delay
lines suitable for the latter case are relatively expensive. In addition, there are concerns about the
amplitude variations with tapped delay lines as different taps are selected, delay uniformity over a
production lot, and delay variations with temperature. In response to these difficulties, there has been
a major migration to digital beam formation over the last 15 years (Thomenius, 1996).
An alternate method of introducing focusing delays for both analog and digital beam formers is
by heterodyning (Maslak, 1979). This is usually done in conjunction with mixing the received signal
with a lower local oscillator frequency with the goal of moving the received energy to a different
location on the frequency spectrum. If the phase of the local oscillator is varied appropriately for
each of the array signals, the location of constructive interference can be placed at a desired location.
The limitations of this are associated with the reduced bandwidth over which the delay correction
will be accurate and the reduced range of phase correction that is possible. Finally, as noted above,
focusing (and beam steering) can be accomplished by relatively straightforward digital techniques in
a digital beam former. A number of different methods of digital beam former implementation have
been published in the sonar and ultrasound literature (Mucci, 1984; Steinberg, 1992).
Figure 9.4 shows the formation of the focal region for a 20-mm aperture circular transducer with
a geometric focal distance of 100 mm and being excited with a CW signal of 3.0 MHz. At the left-
hand side of the pressure profile, the rectangular section from –10 to 10 mm corresponds to the pres-
sure at the transducer aperture. In the near field, there are numerous peaks and valleys corresponding
to areas where there is partial constructive and destructive interference. As one looks closer to the
focal region, these peaks and valleys grow in size as the areas of constructive and destructive inter-
ference become larger. Finally, at the focal point the entire aperture contributes to the formation of
the main beam.
One way of assessing the quality of a beam is to look at its beamwidth along the transducer axis.
The 6- and 20-dB beamwidths are plotted in Fig. 9.5. It is important to recognize that the
beamwidths shown are those for a circular aperture. Due to the axial symmetry, the beamwidths
shown will be achieved in all the planes, that is, in the imaging plane as well as the plane perpen-
dicular to it (this plane is often referred to as the elevation plane from radar literature). This will not
be the case with rectangular transducers. With rectangular transducers, the focusing in the image
plane is done electronically, that is, in a manner similar to annular arrays. However, in the elevation
plane, the focusing in today’s systems is done either by a lens or by the curvature of the elements.
In such cases, the focal location will be fixed and cannot be changed electronically. Remedying this
limitation of rectangular transducers is currently an active area of study. The introduction of the so-
called elevation focusing will be discussed in greater detail in a later chapter.