= Pulsed Doppler depth-velocity tradeoff
The \b[pulsed Doppler depth-velocity tradeoff] is the restriction linking uniquely localized depth and unaliased beam <velocity> in a simple repeated-pulse <Doppler ultrasound> acquisition. Deeper echoes take longer to return, leaving fewer distinct pulse observations per second. Fewer observations reduce the unaliased <Doppler ultrasound> <frequency> interval.
For maximum depth $d$, <speed of sound> $c$ and <pulse repetition frequency> $f_r$, the ideal nonoverlapping timing restriction is $f_r\leq c/(2d)$. The <Nyquist frequency> is $f_r/2$. Combining this with $f_D=2f_0v\cos\theta/c$ gives
$$
|v|<\frac{cf_r}{4f_0|\cos\theta|}
\leq\frac{c^2}{8f_0d|\cos\theta|}.
$$
The strict inequality excludes the ambiguous Nyquist endpoint. For beam-parallel motion, $|\cos\theta|=1$. At a right angle there is no first-order Doppler sensitivity to the motion; the apparent divergence of the formula does not provide unlimited useful <velocity> measurement.
For an original calculation, let $c=1540$ m/s and $f_0=2.5$ MHz. A 15 cm depth allows at most approximately 5133 pulses/s in this idealization, with axial Nyquist <speed> about 0.791 m/s. At 30 cm these bounds fall to approximately 2567 pulses/s and 0.395 m/s. Doubling the transmitted <frequency> to 5 MHz halves the axial <velocity> limits at either depth.
\Image[/pulsed-doppler-depth-velocity-limit.png]
{title=Ideal beam-parallel pulsed Doppler Nyquist speed decreases with depth and transmitted frequency}
These are timing and <sampling (signal processing)> bounds, not a specification for a particular instrument. Pulse duration, scheduling and processing can lower the usable limit. Moving a spectral baseline changes the displayed signed interval but does not increase its total sampled-frequency width. <Continuous-wave Doppler echocardiography> avoids this pulse-rate restriction while sacrificing unique depth selection; high-pulse-rate acquisition instead accepts <Doppler range ambiguity>.<Recommendations for Quantification of Doppler Echocardiography by Miguel A. Quiñones and others>[General Principles, pp. 167–168, PW/CW and high-PRF distinctions]{ref} The acquisition choice determines which ambiguity remains.
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