IMG-18

Doppler shift

Δf / f0 = 2 v cosθ / c. Blood-flow ultrasound.

Reading speed

UltrasoundDoppler

Governing equation

\Delta f=2 f_0 v\cos\theta/c

$v \in \left[1, 250\right]$

where

f_0: Transmit frequency (MHz)
v: Flow speed (cm/s)
\theta: Doppler angle (°)
c: Speed of sound (m/s)
\Delta f: Doppler shift (kHz)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\Delta f=2 f_0 v\cos\theta/c

$v \in \left[1, 250\right]$

$v = 1$ $\Delta f = 0.056$

Along the curve

②

Domain of the variable — blue

v_{\min} = 1,\quad v_{\max} = 250

$v \in \left[1, 250\right]$

$v = 1$ $v = 1$

Along the curve

③

Response curve — green

\Delta f=2 f_0 v\cos\theta/c

$v \in \left[1, 250\right]$

$v = 1$ $\Delta f = 0.056$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta f}{\mathrm{d}v}

$v \in \left[1, 250\right]$

$v = 2.566$ $\Delta f^{\prime} = 0.056$

Along the curve

Lecture brief

Historical brief

Larmor precession, CT Beer projections, SNR and Nyquist sampling are why MRI and CT images exist as numbers. The lab computes frequency, dose and resolution limits. This sheet (IMG-18 — Doppler shift) is the form associated with Doppler. Working symbols: $f_0$ , $v$ , $\theta$ , $c$ $\rightarrow$ $\Delta f$ . The 2 is the round trip. θ = 0 is maximum shift; θ = 90° is blind. Alias when Δf > PRF/2.

Purpose

Purpose: compute

\Delta f

from

f_0

v

\theta

c

in Medical imaging via

\Delta f=2 f_0 v\cos\theta/c

Δf / f0 = 2 v cosθ / c. Blood-flow ultrasound. Use it when a real medical imaging question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f_0 = 5.000\,\mathrm{MHz}

v = 50.000\,\mathrm{cm/s}

\theta = 30.000\,\mathrm{^{\circ}}

c = 1540.000\,\mathrm{m/s}

, the governing relation

\Delta f=2 f_0 v\cos\theta/c

yields

\Delta f = 2.812\,\mathrm{kHz}

. A vessel, an angled beam, a frequency shift. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Transmit frequency f_05 MHzFlow speed v50 cm/sDoppler angle \theta30 °Speed of sound c1540 m/s

Outputs

Doppler shift \Delta f2.812 kHz

Reading speed

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Narration of this film

A vessel, an angled beam, a frequency shift.

The 2 is the round trip. θ = 0 is maximum shift; θ = 90° is blind. Alias when Δf > PRF/2.

Reading speed

Watch on YouTube