Signals and Systems/Frequency Domain Analysis

Frequency Analysis

Noise, like any other signal, can be analyzed using the Fourier Transform and frequency-domain techniques. Some of the basic techniques used on noise (some of which are particular to random signals) are discussed in this section.

Gaussian white noise, one of the most common types of noise used in analysis, has a "flat spectrum". That is, the amplitude of the noise is the same at all frequencies.

Power Spectral Density (PSD) of Gaussian White Noise

White noise has a level magnitude spectrum, and if we square it, it will also have a level Power Spectral Density (PSD) function. The value of this power magnitude is known by the variable N₀. We will use this quantity later.

Wiener Khintchine Einstein Theorem

Using the duality property of the Fourier Transform, the Wiener-Khintchine-Einstein Theorem gives us an easy way to find the PSD for a given signal.

if we have a signal f(t), with autocorrelation R_ff, then we can find the PSD, S_xx by the following function:

$S_{xx}={\mathcal {F}}(R_{ff})$

Where the previous method for obtaining the PSD was to take the Fourier transform of the signal f(t), and then squaring it.

Bandwidth

The bandwidth of a random function.

Windowing

Many random signals are infinite signals, in that they don't have a beginning or an end. To this effect, the only way to really analyze the random signal is take a small chunk of the random signal, called a sample.

Let us say that we have a long random signal, and we only want to analyze a sample. So we take the part that we want, and destroy the part that we don't want. Effectively, what we have done is to multiply the signal with a rectangular pulse. Therefore, the frequency spectrum of our sampled signal will contain frequency components of the noise and the rectangular pulse. It turns out that multiplying a signal by a rectangular pulse is rarely the best way to sample a random signal. It also turns out that there are a number of other windows that can be used instead, to get a good sample of noise, while at the same time introducing very few extraneous frequency components.

Remember duality? multiplication in the time domain (multiplying by your windowing function) becomes convolution in the frequency domain. Effectively, we've taken a very simple problem (getting a sample of information), and created a very difficult problem, the deconvolution of the resultant frequency spectrum. There are a number of different windows that we can use.