# Fourier Transform frequency translation property in practical applications

From the material of my Signal Analysis and Processing course:

The frequency translation property is extremely important for practical applications. It allows taking a signal from a portion of the spectrum and relocating it onto any other portion of the spectrum. Thanks to this property, for instance, radio transmission is possible, including FM radio, cell phones, TV transmission and so on.

I understand this property but I can't find anything on the Internet regarding its practical applications. Do you have links to any resource of this type?

• You might want to read about modulation. Jan 23, 2018 at 9:28
• And the slightly redundant wikipedia article on (equivalent) Baseband signal might be of interest. In short: It doesn't matter what your WiFi's center frequency is. The signal processing done by the WiFi chipset to convert what is originally RF signal to packets of bits (and vice versa) is always the same, thanks to the fact that frequency translation simply is a reversible operation. Jan 23, 2018 at 9:32
– A_A
Feb 15, 2018 at 11:23

I can't find anything on the Internet regarding its practical applications.

Translation, in general, occurs when a series is multiplied with (or "modulated by") a sinusoid. I write "series" because the translation effect can be applied to both the time domain and the frequency domain.

The main thing to keep in mind here is that when you multiply a time series with a sinusoid you "shape" its spectrum in a predictable way.

This is the standard trick used by Heterodyning which mentions a few more applications of the same concept at its "Applications" section.

At the heart of this property are the Product To Sum (and back) trigonometric identities. These point back to the subject of modulation and phase detection too.

But also, this property, along with a similar property that holds for logarithms were behind the proliferation of various types of slide rules which were early aids to support quick calculation. Being able to reduce a complicated calculation involving trigonometric functions down to a set of much simpler additions fuelled rapid expansion in a number of different disciplines in the past. Slide rules used to be found in all sorts of ships.

Hope this helps.

EDIT:

Another variation on the same theme is the modification of the low pass digital filter.

The simplest digital filter everyone is introduced to in DSP is the Sinc Filter. A practical sinc filter, is a low pass filter. By modulating a sinc filter with different types of sinusoids (real or imaginary) your can change its cut-off frequency (in real time) or transform it to any other type of filter (That is, from low pass you can transform it to a high-pass, band-pass or band-reject. One set of coefficients, different modulations).