Lesson 02: Logarithms, dBs, and Power

Outline

  • Logarithms

  • Bels and deciBels

    • Referenced to any power

    • Referenced to sound power

Introduction to Logarithms

Basic Relationships

  • Let us first examine the relationship between the variables of the logarithmic function.
  • The mathematical expression: N=(b)^x
    • Where: b = base, x = exponent

So what’s a logarithm?

  • The inverse of exponents.
  • Specifically, given a number, what is the exponent that the base needs to be raised to get it.
  • In other words, if you know N & b, then what is x?

x = log_b(N)

And why do I care?

  • Doing arithmetic with very large or very small numbers is a pain in the @$$!
  • Try the following without a calculator:
    • 937362478924*21367136314734
    • 0.000215579456123/23148734634
    • 73272089468^836
    • 5√(93648292.92369)
  • Good luck!
  • But with logarithms this becomes a simple chore of addition, subtractions, multiplications, and divisions, respectively.

Properties of Logarithms

There are a few characteristics of logarithms that should be emphasized: (a.k.a. memorized!)

  • The common or natural logarithm of the number 1 is 0

log(1)=0

  • The log of any number less than 1 is a negative number

log(x) < 0, \forall x < 1

  • The log of the product of two numbers is the sum of the logs of the numbers

log(a \cdot b) = log(a) + log(b)

  • The log of the quotient of two numbers is the log of the numerator minus the log of the denominator

log(a \div b) = log(a) - log(b)

  • The log of a number taken to a power is equal to the product of the power and the log of the number

log(x^a) = a \cdot log(x)

Common Implications

  • Your calculator or a program such as MATLAB® may work with functions such as log or ln.
  • By default on a calculator
    • log(x) \rightarrow log_{10}(x)
    • ln(x) \rightarrow log_e(x) (What’s ‘e’ called and what does it represent?)

Examples

Don’t be afraid to use your calculator

  • Log(0.001)
  • Ln(e)
  • Log(e)

Decibels

  • Power Gain

  • Voltage Gain

  • Human Auditory Response

Power Gain

  • Number representing the ratio between two power levels.
  • The basic unit is the Bel (B), defined as: B=log_{10}(P_2/P_1)
  • This value is large, so we work with 1/10th of that, aka deciBel (dB)

dB = 10 \cdot log_{10}(x)

Note: If the new unit is 1/10th of the original, then the value must be made 10 times larger and the net result is essentially multiplying by 1.

  • Common ratios include:

P_2 = 2P_1 \rightarrow \text{Power ratio of } 3 dB

P_2 = 10P_1 \rightarrow \text{Power ratio of } 10 dB

  • Some applications work with a standard power reference at a certain load value, e.g. 600 W.
    • At a 1 mW reference, we have

dBm = 10 \cdot log_{10}(P / 1 mW)

    • At a 1 W reference, we have

dBW = 10 \cdot log_{10}(P / 1 W)

Note: You have to make sure that the base unit inside the logarithm cancels, i.e. Watts. Prefixes should always be taken into consideration seperately, since they stand for exponents, e.g. milli- = 10^{-3}.

Voltage Gain

  • Just as for power gain, we can describe a relative difference between voltage values, a.k.a. voltage gain.
  • Since by definition dBs are a calculation of power ratios, how can we calculate voltage ratios in terms of dB?
  • Let’s start by answering the question: can power be written in terms of voltage?
    • Of course!
    • \text{Power} = \text{Voltage}^2 / \text{Load impedance} (\Omega)
  • Since the power depends on load impedance, calculating for dB becomes a bit more complicated:

\begin{aligned} dB & = 10log_{10}\left(\frac{P_2}{P_1}\right) = 10log_{10}\left(\frac{V^2_2/R_2}{V^2_1/R_1}\right) = 10log_{10}\left(\frac{V^2_2/V^2_1}{R_2/R_1}\right) \\ & = 10log_{10}\left(\frac{V_2}{V_1}\right)^2 - 10log_{10}\left(\frac{R_2}{R_1}\right) \\ & = 20 log_{10}\left(\frac{V_2}{V_1}\right) - 10 log_{10}\left(\frac{R_2}{R_1}\right) \end{aligned}

And what does all of this have to do with telephony?

  • The Bel, though a measure of power ratios, was initially used in Human Auditory Response.
  • The human ear has a frequency response to sound on a logarithmic scale.
  • When working with audio, there is a reference power level of 0.0002 mbar.
    • The resulting dB value with this reference is:

dBs = 20 log_{10}(\text{Pressure} / 0.0002 \text{mbar})

  • To double the sound level received by the human ear, the power rating of the acoustical source (in watts) must be increased by a factor of 10. (Why?)

Other references

  • dBrn = 0 dB referenced to noise.
  • What is the reference for noise? -90 dBm
  • dBrnC0 = dB referenced to noise that passed through a C message filter at a 0 level test point.
    • …..WHAT?!
    • C filter – passes only signals between 300 Hz and 3400 Hz. (The rage of telephony audio.)
  • The dBrnC0 scale starts at a value of 0, but it represent the noise prior to entering the C filter, i.e. -90 dBm.

Signal to noise ratio

  • Ratio of signal power to noise power: SNR = P_2/P_n
  • In dB is becomes: SNR_{dB} = 10log_{10}(P_s/P_n)

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