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I'm still trying to understand the convolution of two functions. Every God damn text ist only about how useful the Fourier series are - that's it. (Math studen

by h3ctic 10y ago

I'm still trying to understand the convolution of two functions. Every God damn text ist only about how useful the Fourier series are - that's it. (Math student)

gajjanag 10y ago
Since you mentioned that you are a math student, I highly recommend Stein and Shakarchi's "Fourier Analysis: An Introduction". It is a rigorous treatment that covers a lot of the essentials of Fourier theory, while also covers some very interesting "math applications" (e.g Dirichlet's theorem on primes in arithmetic progressions at the end), as well as "mixed applications" (e.g Radon transforms that are used in imaging but are also of interest from a math perspective). Note that this book requires an introductory analysis class as a prereq; basically you need to be familiar with the standard Riemann integral theory and rigorous treatment of limits, continuity, and derivatives.
gregfjohnson 10y ago
I similarly struggled to get intuition on the convolution operation, until the following simple idea clicked. Think of convolution as similar to elementary school long multiplication, but without carries. (Also, if you are working with vectors of of some length N, the multiplication "wraps" modular-arithmetic style around the end of the vector back to the beginning.)
jwmerrill 10y ago
It's much easier to get an intuitive understanding of convolution directly in the time domain, without worrying about Fourier Transforms. Convolution with a top-hat kernel is just a moving average. Each point of the output is the average of the input signal over a given radius about that point. Convolution with other kernels is a weighted moving average. Each point of the output is the average of the input signal over a region with weights depending on the displacement from that point. The Fourier Transform allows implementing convolution as multiplication in frequency space (because the Fourier transform turns translations into multiplications), which is sometimes formally useful, and sometimes more efficient to evaluate.

by h3ctic 10y ago

I'm still trying to understand the convolution of two functions. Every God damn text ist only about how useful the Fourier series are - that's it. (Math student)

gajjanag 10y ago

Since you mentioned that you are a math student, I highly recommend Stein and Shakarchi's "Fourier Analysis: An Introduction". It is a rigorous treatment that covers a lot of the essentials of Fourier theory, while also covers some very interesting "math applications" (e.g Dirichlet's theorem on primes in arithmetic progressions at the end), as well as "mixed applications" (e.g Radon transforms that are used in imaging but are also of interest from a math perspective). Note that this book requires an introductory analysis class as a prereq; basically you need to be familiar with the standard Riemann integral theory and rigorous treatment of limits, continuity, and derivatives.

gregfjohnson 10y ago

I similarly struggled to get intuition on the convolution operation, until the following simple idea clicked. Think of convolution as similar to elementary school long multiplication, but without carries. (Also, if you are working with vectors of of some length N, the multiplication "wraps" modular-arithmetic style around the end of the vector back to the beginning.)

jwmerrill 10y ago

It's much easier to get an intuitive understanding of convolution directly in the time domain, without worrying about Fourier Transforms. Convolution with a top-hat kernel is just a moving average. Each point of the output is the average of the input signal over a given radius about that point. Convolution with other kernels is a weighted moving average. Each point of the output is the average of the input signal over a region with weights depending on the displacement from that point. The Fourier Transform allows implementing convolution as multiplication in frequency space (because the Fourier transform turns translations into multiplications), which is sometimes formally useful, and sometimes more efficient to evaluate.