Page 21 - Winter 2008

Fig. 3. Shown are the sampling points of an A/D converter under several scenarios, described in the text.
but then you must add one more black dot at the end to pre- serve the correct total. The piece you added at the end was exactly what you left off to start with. Thus when computing the amplitude of the signal (called detection), for example by multiplying it (mixing it) point by point with a constant- amplitude sine wave of the same frequency and summing the results for one cycle, the sum is independent of the starting point. That is, the result “settles” to its final value in exactly one cycle. This is in sharp contrast to any analog detection and filtering scheme. Here is why. The process of mixing two signals is mathematically identical to multiplication. For two signals of different frequency, the results (a simple trig iden- tity) are the sum and difference of the two frequencies (het- erodyne). If the two signals are of the same frequency as in the RUS numerically-implemented detection scheme described here (homodyne), the result is the sum (2f) and the difference, zero. The amplitude of the signal that we want is contained in either component. This is often obtained by using a low-pass analog or numerically-implemented model of an analog filter to eliminate the 2f component, such as a simple resistor-capacitor (RC). All such filters have some sort of continuous response to frequencies above dc. The conse- quence is that after a step change in amplitude of the input signal, the amplitude of the mixed signal that is reported takes some time to stabilize. For an RC filter set to 2f and a 16-bit digitizer, a full-scale step change in signal amplitude takes about 11 time constants to settle to one least significant bit (1 part in 65536). In contrast, summing one cycle’s worth of data from a SD signal settles in one cycle. Implementing such a scheme in a low-noise system enables one to step through frequencies to find the resonances of the specimen an order of magnitude faster than trying to emulate, or worse, actually use, an analog filter.
In an RUS system, the way SD is implemented is shown in Fig. 4, and a resonance measured with it in Fig. 5. Noting that useful specimens for RUS can be fractions of a millimeter in size with useful resonances up to 10 MHz, and that to obtain good signal-to-noise ratio a 16-or-so bit digitizer is needed to take of order 64 measurements per cycle, it is not a happy thought to try to acquire a 64 mega-sample-per-second 16 bit digitizer. So, our approach is to use a heterodyne mixer before the homodyne numerical detection just described. We gener- ate a frequency f to drive the transmitting transducer, a fre- quency f+Δf to drive an electronic analog multiplier and a fre- quency mΔf to clock a digitizer. By generating a reference sig- nal at Δf as well as the amplified transducer signal, we preserve all phase information. Using 1kHz for Δf, only a 128k samples per second (sps), two-channel digitizer is needed.
Digital process for unambiguous determination of time-of-flight (TOF) in a pulse-echo measurement
Pulse-echo (TOF) ultrasound measurements are widely used, and are thoroughly reviewed by many authors. The basic idea is to launch an acoustic pulse into a specimen and measure the time, t, it takes to reflect from the face of the specimen opposite the transducer. Then, absent many arcane corrections, vs=2l/t where vs is the speed of sound, l is the length of the sample, and only long division is required to process the raw data. A configuration that is in common use
plexity of design while attempting to control cost. For these and other reasons, we (usually) want to input a signal to the A/D converter whose intrinsic noise is greater than the reso- lution of the A/D, and whose bandwidth is less than the Nyquist limit, the lowest “sampling frequency” that can be used. In Fig. 3, a noise-free sinusoidal signal of frequency f is digitized under several scenarios to illustrate the Nyquist- limit problem. Rather than offer a proof, let us find special cases that describe successes and failures in a digital capture of the signal. First we consider the Nyquist limit fN=2f. At this A/D sampling rate it is possible to miss completely the pres- ence of the signal, as shown by the red stars in Fig. 3. Thus 2f is the absolute lower sampling rate bound for digital acquisi- tion of an analog signal. As sampling rate increases above fN (red triangles) we obtain an increasingly accurate digital rep- resentation of the signal. That is, just above fN, every zero crossing of the signal is captured, even if, when we connect the dots, the signal is distorted until we heavily “oversample.” Thus a Fourier transform must contain the fundamental fre- quency in it. Below fN (blue dots) “aliasing” occurs and we obtain a spurious representation that is the difference between an integer multiple of sampling rate and f (this process implements an under-sampling mixer used, among other places, in cell phone radio receivers). More simply, we missed some zero crossings, therefore the Fourier transform of the signal cannot have the actual fundamental frequency in it, only frequencies below f.
The sampling rate that will be discussed and that will be used for RUS is a special one indicated by the black dots in Fig. 3. The goal is to obtain the amplitude and phase of the sinusoidal signal with maximum speed and a useful noise floor, while rejecting other frequencies. The black dots are at a synchronous digitization (SD) rate that samples the signal at an integer multiple m of the fundamental frequency. The advantage of doing this is that for each cycle of the funda- mental, it does not matter where one starts digitizing with respect to the phase of the signal, the set of numbers obtained after an integer multiple of m acquisitions is the same. Try it yourself with Fig. 3. Leave off the first black dot at the start,
Digital Ultrasonics 19