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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive aliasing from sampled complex phases.
- Justify the conclusion "Both sampled cosine sequences are identical" using the stated assumptions.
Before you start
Sinusoids, periodicity, and sample rates.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
At sample rate fₛ=10 Hz, compare sampled cosines of 3 Hz and 7 Hz.
Why this math matters
Derive aliasing from sampled complex phases. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Sampling is uniform and noiseless.
- The waves have the same zero phase and amplitude.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
See how different waves produce identical samples
PausedQuestion: Start with the question. Paused.
Question
Start with the question
At sample rate fₛ=10 Hz, compare sampled cosines of 3 Hz and 7 Hz.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Samples occur at t=n/10
Only integer-indexed sample times are observed.
Work through the mathematics
cos(2π·7n/10)=cos(2πn−2π·3n/10)
Frequencies seven and minus three differ by the sample rate.
Check the conclusion
Both sampled cosine sequences are identical
Cosine is even and 2π-periodic, so the sample data alone cannot distinguish these waves.
The result
Both sampled cosine sequences are identical
Cosine is even and 2π-periodic, so the sample data alone cannot distinguish these waves.
Common mistakes to catch
- The Nyquist condition concerns bandwidth as well as a sample rate.
- Aliasing is not merely small rounding error.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Which lower frequency aliases a 9 Hz cosine at 10 Hz sampling?
Show a hint
Subtract the sampling frequency and use cosine symmetry.
Reveal answer and explanation
1 Hz
Nine is equivalent to minus one modulo ten for these sample phases.
Practice 2
Does increasing numerical precision fix aliasing?
Show a hint
The sequences are exactly equal.
Reveal answer and explanation
No
More bits cannot recover information never distinguished by the sample times.
Take the idea with you
Explain the purpose of a low-pass filter before digitizing an analog signal.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Convert a forced oscillation into a complex amplitude
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