# Why multiplying frequency kernel with sinusoid results shifting frequency domain

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.

• @KadirErdemDemir: a sinusoid at 0.5 frequency is actually like $x[n]=cos(\pi n) = (-1)^n$ whose DTFT is $X(e^{j\omega}) = \pi\delta(\omega-\pi) + \pi\delta(\omega+\pi)$ as klurie states i-since multiplicaiton in time equals convolution in frequency and ii-since convolution with impulse means means shift to impulse location, the result is that lowpass filter which is centered about zero frequency is shifted to impulse frequency at $\omega = \pi$ which is considered as the high frequency region. Hence the filter becomes a highpass filter from lowpass origin. Feb 18, 2015 at 20:28