<< Chapter < Page | Chapter >> Page > |
The concept of decibel originates from telephone engineers who were working with power loss in a telephoneline consisting of cascaded circuits. The power loss in each circuit is the ratio of the power in to the power out, or equivivalently, the power gain isthe ratio of the power out to the power in.
Let ${P}_{\mathrm{in}}$ be the power input to a telephone line and ${P}_{\mathrm{out}}$ the power out. The power gain is then given by
Bel is often a to large quantity, so we define a more useful measure, decibel:
If no reference level is given it is customary to use ${P}_{\mathrm{ref}}=1\; W$ , in which case we have:
Given the power spectrum density (psd) function of a signal $x(n)$ , ${S}_{\mathrm{xx}}(if)$ . Express the magnitude of the psd in decibels.
We find ${S}_{\mathrm{xx}}(\text{dB})=10\lg \left|{S}_{\mathrm{xx}}(if)\right|$ .
Above we’ve calculated the decibel equivalent of power. Power is a quadratic variable, whereas voltageand current are linear variables. This can be seen, for example, from the formulas $P=\frac{V^{2}}{R}$ and $P=I^{2}R$ .
So if we want to find the decibel value of a current or voltage, or more general an amplitude we use:
Express the magnitude of the filter $H(if)$ in dB scale.
The magnitude is given by $\left|H(if)\right|$ ,which gives: $\left|H(\text{dB})\right|=20\lg \left|H(if)\right|$ .
Plots of the magnitude of an example filter $\left|H(if)\right|$ and its decibel equivalent are shown in .
The ratios 1,10,100, 1000 give dB values 0 dB, 10 dB, 20 dB and 30 dB respectively. This implies that an increaseof 10 dB corresponds to a ratio increase by a factor 10.
This can easily be shown: Given a ratio R we have R[dB]= 10 log R. Increasing the ratio by a factor of 10 we have: 10 log (10*R) = 10 log 10 + 10 log R = 10 dB + R dB.
Another important dB-value is 3dB. This comes from the fact that:
An increase by a factor 2 gives: an increase of 10 log 2≈3 dB. A“increase”by a factor 1/2 gives: an“increase”of 10 log 1/2≈-3 dB.
In filter terminology the cut-off frequency is a term that often appears. The cutoff frequency (for lowpass and highpass filters ), ${f}_{c}$ , is the frequency at which the squared magnitude response in dB is½. In decibel scale this corresponds to about -3 dB.
In signal processing we have the following relations for linear systems:
Above we have used ${P}_{\mathrm{ref}}=1\; W$ as a reference and obtained the standard dB measure. In some applications it is moreuseful to use ${P}_{\mathrm{ref}}=1\; mW$ and we then have the dBm measure.
Another example is when calculating the gain of different antennas. Then it is customary to use an isotropic(equal radiation in all directions) antenna as a reference. So for a given antenna we can use the dBi measure. (i ->isotropic)
Notification Switch
Would you like to follow the 'Information and signal theory' conversation and receive update notifications?