Noise performance characterization

In this section, we will describe some quantities that are used for the description of the noise behavior of information processing systems.

Signal-to-noise ratio

The signal-to-noise ratio \(SNR\), of a signal perturbed by noise is defined as the ratio of the signal power and the noise power , and is often interpreted as a quality measure for the signal:

\[\begin{split}SNR & =\frac{P_{signal}}{P_{noise}}~\text{[-],}\\ \left( SNR\right) _{\text{dB}} & =10\log_{10}\left( \frac{P_{signal} }{P_{noise}}\right) ~\text{[dB].}\end{split}\]

Dynamic range

A quality measure for amplifiers that is closely related to the channel capacity is the dynamic range \(D\) of an amplifier. It is defined as the ratio of the maximum signal power \(P_{signal,\max}\) at which the retrieval of the information is still possible, and the noise power \(P_{noise,\min}\) in absence of a signal:

\[D=10\log_{10}\left( \frac{P_{signal,\max}}{P_{noise,\min}}\right) ~\text{[dB].}\]

The maximum power \(P_{signal,\max}\) that is used in this definition differs for each application. The Intermodulation-Free Dynamic Range \(IMFDR\) of a low-noise RF amplifier is based on the output power level where the power of the intermodulation components equals that of the noise. In audio amplifiers, the dynamic range is measured at an output power for a given percentage of total harmonic distortion.

Noise figure

The noise figure \(NF\) [dB] or \(F\) [-] of an amplifier tells us something about the deterioration of the signal-to-noise ratio by the amplifier. \(NF\) and \(F\) are defined as:

\[\begin{split}NF & =10\log F,\\ F & =\frac{S/N\text{ {\small at the input of the amplifier}}}{S/N\text{{\small at the output of the amplifier}}}.\end{split}\]

The noise figure is usually defined at a temperature of \(290\)K.

Equivalent noise bandwidth

The equivalent noise bandwidth \(B_{n}\) of a system with a transfer function \(H(j\omega)\) is defined as the bandwidth of a brickwall filter with a pass band gain equal to the maximum magnitude of \(H(j\omega)\), that would produce the same output noise power as \(H(j\omega)\):

\[B_{n}=\frac{1}{2\pi}\int_{0}^{\infty}\left\vert \frac{H(j\omega)}{H_{\max} }\right\vert ^{2}d\omega~\text{[Hz].}\]

Example

We will evaluate the noise bandwidth of a first order low-pass filter of which the transfer \(H(j\omega)\) can be written as

\[H(j\omega)=\frac{H_{0}}{1+j\omega\tau}.\]

The maximum value of \(\left\vert H(j\omega)\right\vert \) is \(H_{0}\), from which we obtain

\[\begin{split}B_{n} & =\frac{1}{2\pi}\int_{0}^{\infty}\left\vert \frac{1}{1+j\omega\tau }\right\vert ^{2}d\omega\\ & =\frac{1}{2\pi\tau}\int_{0}^{\infty}\frac{1}{1+\omega^{2}\tau^{2}} d\omega\tau\nonumber\\ & =\frac{1}{4\tau}\\ & =\frac{\pi}{2}B_{-3\text{dB}}.\end{split}\]

Hence, the noise bandwidth of a first order low-pass filter is \(\pi/2\) times larger than its \(-3\)dB bandwidth in Hz.