Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute a small discrete Fourier transform with an explicit sign convention.
- Justify the conclusion "X=(0,2,0,2)" using the stated assumptions.
Before you start
Complex numbers and Euler's formula.
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
For x=(1,0,−1,0), compute Xₖ=Σₙ₌₀³xₙexp(−2πikn/4).
Why this math matters
Compute a small discrete Fourier transform with an explicit sign convention. 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:
- Indices n and k run from zero to three.
- The forward transform uses the negative exponential sign.
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.
Resolve four samples into discrete frequency components
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For x=(1,0,−1,0), compute Xₖ=Σₙ₌₀³xₙexp(−2πikn/4).
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
Only n=0 and n=2 contribute, so Xₖ=1−exp(−πik)
Zero samples remove two terms from each sum.
Work through the mathematics
exp(−πik)=(−1)ᵏ
Integer-frequency phase simplifies to alternating signs.
Check the conclusion
X=(0,2,0,2)
The two conjugate frequency bins represent the sampled real cosine, under this unnormalized forward-transform convention.
The result
X=(0,2,0,2)
The two conjugate frequency bins represent the sampled real cosine, under this unnormalized forward-transform convention.
Common mistakes to catch
- Different DFT normalization conventions change coefficient sizes.
- The fourth bin is not an unrelated positive-frequency copy for real input.
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
What is the transform of (1,1,1,1)?
Show a hint
Separate the constant mode.
Reveal answer and explanation
(4,0,0,0)
Roots of unity cancel in nonzero frequency bins.
Practice 2
What factor is needed in the inverse under this convention?
Show a hint
Forward sums were not normalized.
Reveal answer and explanation
1/4
The inverse must divide by the number of samples to recover the original values.
Take the idea with you
Explain why a constant brightness profile appears only in the zero-frequency bin.
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: See how different waves produce identical samples
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