Building high frequencies out of low frequencies

If you have sines and cosines of some fundamental frequency, and you’re able to take products and sums, then you can construct sines and cosines at any multiple of the fundamental frequency.

Here’s a proof.

\begin{align*} \cos n\theta + i\sin n\theta &= \exp(in\theta) \\ &= \exp(i\theta)^n \\ &= (\cos\theta + i\sin\theta)^n \\ &= \sum_{j=0}^n {n\choose j} i^j(\cos\theta)^{n-j} (\sin\theta)^j \end{align*}

Taking real parts gives us cos nθ in the first equation and the even terms of the sum in the last equation.

Taking imaginary parts gives us sin nθ in the first equation and the odd terms of the sum in the last equation.

One thought on “Building high frequencies out of low frequencies

  1. .. and, revisiting an earlier post of yours a bit, if you take the real part of both sides and note the sine terms are to an even power (which can be easily re-written in terms of cosine without square roots), you’ll notice cos n\theta is a polynomial function of cos \theta. Presto! An expression for Chebyshev polynomials (of the first kind). You can also get the Chebyshev polynomials of the second kind from the imaginary part of the expression, but it’s not quite as pretty.

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