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@ -349,3 +349,9 @@ http://dx.doi.org/10.1137/S1052623499362822) — I used the "linear and separabl
* [Fast Fourier transforms (FFTs)](https://en.wikipedia.org/wiki/Fast_Fourier_transform): DFTs (and many related problems) in O(N log N) operations. Derived the [Cooley–Tukey FFT algorithm](https://en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm) and mentioned a few other algorithms. See [slides](https://github.com/mitmath/18335/blob/spring21/notes/FFT.pdf).
**Further reading**: Textbook sections IV.1–IV.2 and [OCW lecture 31](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-31-eigenvectors-of-circulant-matrices-fourier-matrix/) and [lecture 32](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-32-imagenet-is-a-cnn-the-convolution-rule/). The [Wikipedia FFT article](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (partially written by SGJ) was still not bad last I checked. [Gauss and the history of the fast Fourier transform](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.309.181) (1985) is a wonderful article on the historical development of the FFT. [Duhamel & Vetterli (1990)](https://doi.org/10.1016%2F0165-1684%2890%2990158-U) is a classic review article. SGJ co-developed a little FFT library called [FFTW](https://www.fftw.org/).
## Lecture 32 (Apr 25)
Fourier series vs. DFT: If we view the DFT as a [Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum) approximation for a [Fourier series](https://en.wikipedia.org/wiki/Fourier_series) coefficient (which turns out to be *exponentially* accurate for smooth periodic f(t)!), then the errors are an instance of [aliasing](https://en.wikipedia.org/wiki/Aliasing) (see e.g. the ["wagon-wheel" effect](https://en.wikipedia.org/wiki/Wagon-wheel_effect)). For band-limited signals where we sample at a rate > twice the bandwidth, there is no aliasing and no information loss, a result known as the [Nyquist—Shannon sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem); in the common case where the bandwidth is centered at ω=0, this corresponds to sampling at more than *twice* the highest frequency.
**Further reading**: For a periodic function, a Riemann sum is equivalent to a trapezoidal rule (since the 0th and Nth samples are identical), and the exponential convergence to the integral is reviewed by [Trefethen and Weideman (2014)](https://epubs.siam.org/doi/pdf/10.1137/130932132); SGJ gave a [simplified review for IAP (2011)](https://math.mit.edu/~stevenj/trap-iap-2011.pdf). The subject of aliasing, sampling, and signal processing leads to the field of [digital signal processing (DSP)](https://en.wikipedia.org/wiki/Digital_signal_processing), on which there are many books and courses. A classic textbook is [*Discrete-Time Signal Processing*](https://research.iaun.ac.ir/pd/naghsh/pdfs/UploadFile_2230.pdf) by Oppenheim and Schafer, and there are whole courses at MIT (like 6.3000 and 6.7000) on these topics.