0.10 Models of noise  (Page 3/3)

Application: Quantization In many application, such as inference in wireless communication and such as in quantization insignal processing, a commonly accepted model of the effect of a source of errors on the signal $x\left(t\right)$ is the so-called additive white noise model :

For a quantization with precision $\Delta$ the error is the quantification error ${\epsilon }_k=y_k-x_k$ and it is determined by the signal itself. [One could argue, that ${\epsilon }_k$ is not random since it is completely determined once $x_k$ is known. Still, the model is useful since we can usually not predict ${\epsilon }_{k+1}$ from observing $x_1,..,x_k$ .]

For quantization using rounding (matlab: round The error ${\epsilon }_k$ is in this case uniformly distributed on the interval $[-\Delta /2,\Delta /2]$ . ) it can be shown that the noise power per sampleamounts to $P_{\mathrm{Quant}.\mathrm{noise}}={\sigma }^2=\frac{{\Delta }^2}{12}$ . When using one more bit for quantization, then the error $\Delta$ is half as large, thus the power 4 times smaller, which amounts to roughly -6 dB. In otherwords, the power of the quantization noise is proportional to -6dB times the number of bits used.

Using the analog of Parseval's equation and recalling that $S\left(f\right)$ is the power spectrum (analog of the square of the Fourier transform) we have indeed:

For $K$ samples of noise taken over a time interval of length $K/f_e$ we have, thus, approximatively

Note that the FFT increases power by $K$ . The relation [link] becomes exact when taking expected values $\mathrm{I}\mathrm{E}$ . The approximation improves the larger $K$ is, since the left side is an estimator of the variance ${\sigma }^2$ , which is ${\Delta }^2/12$ for quantization with precision $\Delta$ .

Application: Interference A further example of a situation where an additive white noise proves useful is wireless transmissionof a binary signal under interference. Here, bits may flip from 0 to 1 and vice versa since detection is not perfect. The error ${\epsilon }_k=y_k-x_k$ is here determined by the interfering signal, and the configuration of the decoder. Note that the possible values of each ${\epsilon }_k$ is either 0 (no flip) or 1 (flip). Without specific information on the interference, the chance of a flip is independentof the time $k$ , and independent of the past occurrence of flips. Thus, white noise is a very reasonable model for ${\epsilon }_k$ . Clearly, the probability $P[{\epsilon }_k=0]$ will be close to 1 if only little inference is present and will decrease the stronger the inference.

Gaussian noise As an important special case we mention Gaussian white noise, where the common distribution of the ${\epsilon }_k$ is Gaussian, or “normal”: This model assumption is standard whenevernothing is known on the distribution. It makes sense, e.g., as model for an overall error which is composed of several small unknownerrors. (compare Central Limit Theorem)

Colored Noise A sequence $(n_1,n_2,n_3,...)$ is called colored noise if its terms $n_k$ are random and possess a relation or dependence between them. Consequently, the powerspectrum of colored noise is not flat, but possesses certain prevalent frequencies — hence the name “colored” (recall thatthe frequencies of light waves correspond to colors).

One of the most simple ways to produce colored noise is to filter white noise. For instance, $m_k={\epsilon }_k+{\epsilon }_{k+1}$ is colored since the entries $m_k$ are no longer independent: $m_0={\epsilon }_0+{\epsilon }_1$ and $m_1={\epsilon }_1+{\epsilon }_2$ contain the same number ${\epsilon }_1$ as an additive term. Similarly, $n_k={\epsilon }_k-{\epsilon }_{k+1}$ is colored.

Adopting a continuous-time notation (for convenience) we write

with Fourier transform

and similarly

and power spectrum The same formula for ${\left|M\left(f\right)\right|}^2$ and ${\left|N\left(f\right)\right|}^2$ could be obtained by computing it as the Fourier transform of the auto-correlation (see [link] ); indeed, for $m\left(t\right)$ : $r\left(0\right)=2{\sigma }^2$ , $r\left(1\right)=r(-1)={\sigma }^2$ and $r\left(k\right)=0$ for all other $k$ ; for $n\left(t\right)$ the same except $r\left(1\right)=r(-1)=-{\sigma }^2$ . , (use that $2{cos}^2\left(x\right)=1+cos\left(2x\right)$ ) and $2{sin}^2\left(x\right)=1-cos\left(2x\right)$ )

where ${\sigma }^2$ is the total power of the original noise ${\epsilon }_k$ and $S\left(f\right)={\left|E\left(f\right)\right|}^2={\sigma }^2$ . Note that neither ${\left|M\left(f\right)\right|}^2$ nor ${\left|N\left(f\right)\right|}^2$ are flat. Verification via matlab is easy.