I am having great time while reading "The Scientist and Engineer's Guide to Digital Signal Processing" book.
In "Chapter-14 Introduction to Filters" author indicates we can create high pass filter from low pass filters by multiplying low pass filter's frequency kernel with a sinusodial which has 0.5 frequency. Link to the this chapter
I thought I read every chapter carefully but I can't understand why this will work? Can somebody please explain the reasoning behind of this process as easy that a computer science graduate could understand.
Answer
This can be explained by the Convolution theorem
As you state, in the time domain you have the kernel (f) multiplied by a sinusoid (g).
If we take the Fourier transform of f·g (i.e., F(f·g)), this is equivalent to the convolution between F(f) and F(g) (i.e., F(f)*F(g)). Note that the Fourier transform of the sinusoid is proportional to two delta functions located at the frequency and negative frequency of the sine wave. So when you convolve these deltas with the frequency kernel your frequency kernel "shifts" to be centered around the sinusoid frequency.
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