# Biomedical signals: filtering cannot undo an ambiguous sampling step — Study worksheet

[Study guides](https://unireadinglist.com/guides/biomedical-signal-processing/)

[Course topic: Biomedical Signal Processing](https://unireadinglist.com/subjects/biomedical-engineering/#course-biomedical-signal-processing)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

An invented discrete-time signal and an idealized analog acquisition comparison, not a patient waveform or clinical detector. Smith is a freely readable general DSP anchor with limited copyright permissions; Rangayyan and Krishnan add biomedical reasoning; Semmlow and Griffel connect implementation and imaging.

## Choose your starting point

- [ ] I need freely readable DSP foundations
  Use Smith for sampling, convolution and moving-average filters.

- [ ] I need biomedical analysis decisions
  Choose Rangayyan and Krishnan after signals, probability and basic physiology.

- [ ] I want numerical implementation
  Choose Semmlow and Griffel with calculus, signals and MATLAB or equivalent coding skills.

## Compare the three books

### The Scientist and Engineer's Guide to Digital Signal Processing

Steven W. Smith · 1997
General DSP foundation

Choose it when: You can manipulate sinusoids, sums and indexed sequences.
Use another route when: You require a biomedical validation protocol or assume free reading means unrestricted redistribution.
Selected reading: Sampling, convolution and Moving Average Filters; connect the time and frequency views.
Limits and trade-offs: Accessible explanation is useful before specialized methods, but it is not a clinical performance study or a CC-licensed work.

#### Try before choosing

Evaluate x\[−1\] before filtering the first displayed sample in the main case.
Check your work: The all-integer formula gives x\[−1\]=0; zero padding is not the stated history.

Your notes:

________________________________

ISBN: 9780966017632

[Official free reading](https://www.dspguide.com/pdfbook.htm)

[Book record and access options](https://unireadinglist.com/subjects/biomedical-engineering/scientist-and-engineers-guide-to-digital-signal-processing/)

### Biomedical Signal Analysis

Rangaraj M. Rangayyan; Sridhar Krishnan · 3 · 2024
Biomedical analytical spine

Choose it when: You know linear systems, probability and physiological signal origins.
Use another route when: You need a first introduction to algebra or a proven diagnosis from a toy example.
Selected reading: Biosignal origins, concurrent/coupled processes and filtering for artifact removal before event detection.
Limits and trade-offs: Biomedical examples motivate choices, but a filter’s mathematical response does not itself validate an endpoint or diagnosis.

#### Try before choosing

For h=\[1/2,1/2\], find the response to the alternating sequence (−1)^n.
Check your work: Adjacent samples cancel exactly; H(π)=0.

Your notes:

________________________________

ISBN: 978-1-119-82585-2

[Book record and access options](https://unireadinglist.com/subjects/biomedical-engineering/biomedical-signal-analysis-rangayyan/)

### Biosignal and Medical Image Processing

John L. Semmlow; Benjamin Griffel · 3 · 2014
Implementation and image route

Choose it when: You can translate equations into code and inspect sampling assumptions.
Use another route when: You only want conceptual prose or software to choose the acquisition design.
Selected reading: Signal measurement, sampling, discrete systems and filtering before later medical-image material.
Limits and trade-offs: Coding exposes boundary choices and numerical errors, while the later imaging scope adds reading load; software access is separate.

#### Try before choosing

Compare sin(2π50t) and sin(2π150t) sampled at 200 Hz.
Check your work: At t=n/200 they are opposite sequences, so synchronized equal-amplitude terms cancel.

Your notes:

________________________________

ISBN: 9781466567368

[Book record and access options](https://unireadinglist.com/subjects/biomedical-engineering/biosignal-and-medical-image-processing/)

## Check your preparation

### Is the sequence finite or only the display finite?

Your notes:

________________________________

### What does the two-tap average do to a sinusoid?

Your notes:

________________________________

### Where is anti-alias filtering applied?

Your notes:

________________________________

## A six-step study plan

- [ ] Declare the history

Selected reading: Sequences and convolution.

What to do: Write x\[n\] for all integers and list x\[−1\] through x\[3\].

Check your work: The list is 0,3,2,3,0.

Your notes:

________________________________

- [ ] Convolve directly

Selected reading: Moving Average Filters.

What to do: Compute y\[0\] through y\[3\] by hand or exact arithmetic.

Check your work: 1.5,2.5,2.5,1.5.

Your notes:

________________________________

- [ ] Resolve components

Selected reading: Frequency response.

What to do: Calculate effects on DC, alternating term and desired sinusoid separately.

Check your work: DC passes, alternation vanishes, the sinusoid attenuates and shifts.

Your notes:

________________________________

- [ ] Check the output formula

Selected reading: Linear systems.

What to do: Evaluate the closed-form output at the four displayed indices.

Check your work: It agrees with direct convolution, including n=0.

Your notes:

________________________________

- [ ] Change acquisition

Selected reading: Sampling and the follow-up.

What to do: Sample the two specified analog sines and derive their cancellation.

Check your work: The unfiltered acquired sequence is exactly constant 2.

Your notes:

________________________________

- [ ] Place the preventive filter

Selected reading: Acquisition reasoning.

What to do: Apply the stated ideal analog low-pass before sampling and then the same digital filter.

Check your work: Preventing alias cancellation is different from recovering it afterward.

Your notes:

________________________________

## Worked exercise: A moving average with a specified infinite history

For every integer n let x\[n\]=2+cos(πn)+sin(πn/2). Apply y\[n\]=(x\[n\]+x\[n−1\])/2. Give a closed form, the samples n=0,…,3 and the effect on each component.

Assumptions: Exact arithmetic, a linear time-invariant two-tap filter and the given all-integer history. No startup reset, zero padding, finite-window truncation or patient measurement is implied.

Your notes:

________________________________

## Change an assumption: Change the analog acquisition and expose alias cancellation

Instead sample xa(t)=2+sin(2π50t)+sin(2π150t) at fs=200 Hz, synchronously at t=n/fs with the stated common phase origin. Compare no analog filter with an ideal unit-gain brick-wall low-pass passing |f|\<90 Hz before the ADC. Apply the same two-tap digital filter afterward.

Your notes:

________________________________

## Before moving on

- Convolve with the actual input history.
- Explain amplitude and phase changes component by component.
- Demonstrate alias ambiguity and the role of pre-sampling filtering.

## Answer and interpretation

### Is the sequence finite or only the display finite?

Here x\[n\] is defined for every integer. Only n=0,…,3 is displayed; the filter can use x\[−1\].

### What does the two-tap average do to a sinusoid?

H(ω)=(1+e^(−iω))/2=e^(−iω/2)cos(ω/2). At π/2 it attenuates by 1/√2 and delays phase by π/4.

### Where is anti-alias filtering applied?

To the analog signal before sampling. A digital filter after an ambiguous sampling step cannot identify the lost analog components.

### A moving average with a specified infinite history

1. The constant 2 is unchanged. Consecutive cos(πn) terms are opposite, so the alternating component vanishes.
2. The sine average is \[sin(πn/2)+sin(πn/2−π/2)\]/2=(1/√2)sin(πn/2−π/4).
3. Thus y\[n\]=2+(1/√2)sin(πn/2−π/4). At n=0,1,2,3 this is 1.5,2.5,2.5,1.5. In particular x\[−1\]=0, so y\[0\]=(3+0)/2.

DC passes and alternation is removed, while the π/2 sinusoid is attenuated by 1/√2 and phase-shifted by −π/4, corresponding to half a sample.

## A moving average with a specified infinite history — python

```python
from fractions import Fraction

def x(n):
    return (3, 2, 3, 0)[n % 4]

y = [Fraction(x(n) + x(n - 1), 2) for n in range(4)]
print([float(v) for v in y])
# [1.5, 2.5, 2.5, 1.5]
```

Test the result: H(0)=1,H(π)=0,H(π/2)=(1−i)/2. These frequency responses reproduce the direct result; removing a nuisance term did not preserve the desired component exactly.

### Change an assumption: Change the analog acquisition and expose alias cancellation

1. Without analog filtering, the terms become sin(πn/2) and sin(3πn/2)=−sin(πn/2) for all integer n. Hence x\[n\]=2 and the digital output is 2.
2. The specified analog low-pass passes DC and 50 Hz, and removes 150 Hz before sampling. Its acquired sequence is 2+sin(πn/2).
3. The same digital average then yields 2+(1/√2)sin(πn/2−π/4). Its four displayed samples are again 1.5,2.5,2.5,1.5, but now for a different input and acquisition model.

Unfiltered acquisition gives a constant output. Ideal pre-ADC filtering preserves a sampled 50 Hz component, which the digital average still attenuates and delays.

Test the result: The unfiltered constant samples are also compatible with a purely constant analog input; no post-sampling algorithm can distinguish those two specified sources from these samples alone. The ideal analog filter prevents this cancellation rather than recovering information afterward.

## Common mistakes

### Unstated startup history

Zero padding changes initial outputs when the model specifies an ongoing signal.

### Smoothing preserves the wanted waveform

Amplitude and phase response must both be checked.

### Digital filtering repairs every alias

Distinct analog inputs can have identical sampled sequences.

## Sources and edition checks

- [Author: DSP guide and editions](https://www.dspguide.com/editions.htm)
  The author distinguishes the selected 1997 hardcover from the later paperback and sets specific free-reading permissions.

- [Wiley: third-edition Biomedical Signal Analysis](https://onlinelibrary.wiley.com/doi/book/10.1002/9781119825883)
  The third edition covers biosignal origins, coupled processes, artifact filtering and later detection methods.

- [CRC: exact third-edition hardback](https://www.routledge.com/Biosignal-and-Medical-Image-Processing/Semmlow-Griffel/p/book/9781466567368)
  The selected third edition combines signal-processing implementation with later medical imaging; it does not supply clinical validity for this invented signal.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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