Signals and Systems/Frequency Domain Analysis

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Frequency Analysis[edit | edit source]

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.

Stationary vs Ergodic Functions[edit | edit source]

Power Spectral Density (PSD) of Gaussian White Noise[edit | edit source]

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[edit | edit source]

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 Rff, then we can find the PSD, Sxx by the following function:

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

Bandwidth[edit | edit source]

The bandwidth of a random function.

Noise-Equivalent Bandwidth[edit | edit source]

Band limited Systems[edit | edit source]

Narrow band Systems[edit | edit source]

Windowing[edit | edit source]

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.

Triangular Window[edit | edit source]

Hamming Window[edit | edit source]