SME
Bioinstrumentation · Problem type
Sampling fast enough
Use this for “What is the slowest rate that samples a 150 Hz ECG without aliasing?”
Diagram
Alias: |150 − 200 × 1| = 50 Hz
This is an example. Type your own values to start a new problem.
fR (Hz)
=
2
×
fₘₐₓ (Hz)
fN (Hz)
=
fₛ (Hz)
2
fₐ
Alias frequency
worked out (not typed) · Hz
Formulas
fR = 2 × fₘₐₓ
300 = 2 × 150
fN
=
fₛ
2
100
=
200
2
fₐ
=
|fₘₐₓ
−
fₛ
×
round(
fₘₐₓ
fₛ
)|
50
=
|150
−
200
×
round(
150
200
)|
Assumptions
•
The signal holds nothing above fₘₐₓ; a pure tone at fₘₐₓ is the hardest part to keep.
•
Sampling faster than fR, twice fₘₐₓ (the Nyquist rate), keeps every part; fN = fₛ ÷ 2 (the Nyquist frequency) is the highest the samples show.
•
Sampled too slowly, a tone shows up at its alias: its distance to the nearest whole multiple of fₛ.
Step-by-step
WE KNOW
fₘₐₓ = 150 Hz, fₛ = 200 Hz
FIND
slowest safe rate (fR), highest frequency shown (fN), alias frequency (fₐ)
Step 1 · Find slowest safe rate (fR)
Use fR = 2 × fₘₐₓ
Two samples a cycle of the fastest part: sample faster than this.
fR = 2 × 150
fR = 300 Hz
Step 2 · Find highest frequency shown (fN)
Use fN = fₛ ÷ 2
The highest frequency the samples can show.
fN
=
200
2
fN = 100 Hz
Step 3 · Find alias frequency (fₐ)
Use fₐ = |fₘₐₓ − fₛ × round(fₘₐₓ ÷ fₛ)|
Take away the nearest whole number of fₛ: what is left is what the samples show.
fₐ
=
|150
−
200
×
round(
150
200
)|
fₐ = 50 Hz
Check
Put the numbers back in. Both sides should match.
300 = 2 × 150 ✓
100
=
200
2
✓
50
=
|150
−
200
×
round(
150
200
)|
✓
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