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Chatbot math/Bard/24.02/Teaching the discrete Fourier transform to beginners

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Asking Bard for ideas on introducing the discrete fourier transform at the most elementary level

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What astonishes me about this conversation with Bard was the vagueness of my question. Bard not only seems to understand exactly what I wanted, but went out of his/her/their way to accommodate me. My guess is that the humans who created Bard shared my love of mathematics as they fine-tuned Bard before it went public. Guy vandegrift (discuss • contribs) 23:39, 1 February 2024 (UTC)

MY QUESTION:

A fundamental formula establishing the discrete fourier transform is a relationship about the sum of exponentials raised to a complex number. I want to do this with sine waves only and need the corresponding identity for sin(nx/N) sin(mx/N). Can you help me?

BARD'S RESPONSE:

You're on the right track! While the full discrete Fourier Transform uses complex exponentials, you can indeed derive an identity involving sines that captures the essence of its connection to sums of sines. Here's the derivation:

Starting Point:

We want to analyze the product of two sine waves with frequencies m and n, scaled by the period N and represented by complex exponentials with imaginary unit i: 

sin(mx/N) sin(nx/N) = (e^(imπx/N) - e^(-imπx/N)) / 2i * (e^(inπx/N) - e^(-inπx/N)) / 2i

Simplifying:

Using the product-to-sum and Euler's identities, we can expand and rearrange terms: 

= (e^(i(m+n)πx/N) + e^(i(m-n)πx/N) - e^(-i(m+n)πx/N) - e^(-i(m-n)πx/N)) / (4 * 2^2)
= (cos((m+n)πx/N) + i sin((m+n)πx/N) + cos((m-n)πx/N) - i sin((m-n)πx/N)) / 16

Extracting Sines:

Taking the real and imaginary parts separately, we obtain the identities for sines:

Real part: 

cos((m+n)πx/N) cos((m-n)πx/N) / 8 = (sin(nx/N) sin(mx/N) + sin((m+2n)πx/N) sin((m-2n)πx/N)) / 4

Imaginary part: 

sin((m+n)πx/N) cos((m-n)πx/N) / 8 = (sin(mx/N) cos(nx/N) - sin((m+2n)πx/N) cos((m-2n)πx/N)) / 4

These identities relate the product of two sines at frequencies m and n to the sum of products of sines at various combinations of those frequencies, all scaled by the period N. This reflects the essence of the discrete Fourier Transform, where a signal composed of various frequencies can be represented as a sum of sine waves.

Important notes:

  • These identities hold true for integers m, n, and N, where N > 0.
  • The terms on the right-hand side represent a convolution in the frequency domain, which is a fundamental concept in signal processing.
  • These identities are not directly equivalent to the full discrete Fourier Transform, which uses complex exponentials for more compact representation and efficient computation.

I hope this helps!

Question: I know of a convincing plausibility argument for this identity if we use Euler's equation. Is there a corresponding simple image for real-valued functions?
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BARD'S RESPONSE:

While there isn't a single, direct visual equivalent for the identities involving real-valued functions, there are ways to visualize the concept using multiple interrelated graphs:

1. Visualization with Phasors:

  • Imagine two sine waves with frequencies m and n as represented by phasors (rotating arrows) in the complex plane.
  • Multiplying the phasors geometrically corresponds to adding their angles and scaling the length.
  • Visualize the projections of the resulting phasor onto the real and imaginary axes. These projections map to the terms on the right-hand side of the identities.
  • Overlaying multiple pairs of phasors and their projections can build intuition for how different frequency combinations contribute to the product.

2. Time-domain plots:

  • Plot the individual sine waves sin(mx/N) and sin(nx/N) over time.
  • Plot the right-hand side of the identities, considering each term separately. These represent sums of sine waves with shifted frequencies.
  • Observe how the product oscillates in a way influenced by both individual frequencies and their combinations.
  • Animate the plots to see how the product evolves over time and how it relates to the individual components.

3. Frequency-domain plots:

  • Plot the frequency spectrum of the individual sine waves, showing their amplitude at different frequencies.
  • Plot the frequency spectrum of the right-hand side of the identities, which will show peaks at various combinations of the original frequencies.
  • This demonstrates how the product distributes its energy across different frequency components.

4. Convolution visualization:

  • Represent the individual sine waves by their "impulse responses" in the frequency domain (functions with peaks at their corresponding frequencies).
  • Convolve these functions graphically or computationally. The result will resemble the frequency spectrum of the product.
  • This visually depicts how the convolution operation in the frequency domain relates to the product in the time domain.

Remember, these are not perfect substitutes for the complex-valued identities, but they can provide an intuitive understanding of the underlying concepts when combined with mathematical reasoning and experimentation.

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