Linearity
Contents
6.1. Linearity#
The first major property of the DFT that we’ll cover is linearity.
We’ve already seen linearity in the context of LSI systems, where we used it to understand what happens when you convolve a filter
Here, we’ll use linearity slightly differently, but the basic idea is the same.
Property 6.1 (DFT Linearity)
For any pair of signals
We’ll prove this property algebraically, starting from the definition of the DFT (5.12):
Proof. Let
So if we scale and mix two input signals, the resulting DFT component is the same scaling and mixing of their individual DFT components
What does linearity buy us?
DFT linearity is important because most interesting signals are not just simple sinusoids.
What DFT linearity says is that if we can represent an arbitrary signal
As we will see in the next chapter (and as we’ve hinted at earlier), it turns out that all signals can be represented as a weighted combination of sinusoids.
6.1.1. Magnitude is not linear#
Note that DFT linearity applies to the complex numbers
However, it is a common mistake to add magnitudes rather than the full complex numbers.
Magnitude is not linear.
If signals
But remember,
Example
For an extreme case, consider
If we were to add the magnitude spectra, we’d get
However, if we mix the two signals in the time domain, we get the empty signal:
The empty signal has an empty spectrum:
in general.
What this says is that mixing signals does not equate to mixing DFT magnitudes.
Fig. 6.1 Complex numbers