Table of Contents

- 1 Why use decibels in our calculations?
- 2 Definition of dB
- 3 Conversion from decibels to percentage and vice versa
- 4 Using dB values in computations
- 5 Adding voltages
- 6 What do we measure in decibels?
- 6.1 Signal-to-noise ratio (S/N)
- 6.2 Noise
- 6.3 Averaging noise signals
- 6.4 Noise factor, noise figure
- 6.5 Phase noise
- 6.6 S parameters
- 6.7 VSWR and reflection coefficient
- 6.8 Field strength
- 6.9 Antenna gain
- 6.10 Crest factor
- 6.11 Channel power and adjacent channel power
- 6.12 Modulation quality EVM
- 6.13 Dynamic range of A/D and D/A converters
- 6.14 dB (FS) (Full Scale)

- 7 Few numbers worth knowing
- 8 Smartphone Apps

# Why use decibels in our calculations?

Engineers have to deal with numbers on an everyday basis, and some of these numbers can be very large or very small. In most cases, what is most important is the ratio of two quantities. For example, a mobile radio base station might transmit approx. 80 W of power (antenna gain included). The mobile phone receives only about 0.000 000 002 W, which is 0.000 000 002 5 % of the transmitted power. Whenever we must deal with large numerical ranges, it is convenient to use the logarithm of the numbers. For example, the base station in our example transmits at +49 dBm while the mobile phone receives -57 dBm, producing a level difference of +49 dBm – (-57 dBm) = 106 dB. Another example: If we cascade two amplifiers with power gains of 12 and 16, respectively, we obtain a total gain of 12 times 16 = 192 (which you can hopefully calculate in your head – do you?). In logarithmic terms, the two amplifiers have gains of 10.8 dB and 12 dB, respectively, producing a total gain of 22.8 dB, which is definitely easier to calculate. When expressed in decibels, we can see that the values are a lot easier to manipulate. It is a lot easier to add and subtract decibel values in your head than it is to multiply or divide linear values. This is the main reason we like to make our computations in decibels.

# Definition of dB

Although the base 10 logarithm of the ratio of two power values is a dimensionless quantity, it has units of “Bel” in honor of the inventor of the telephone (Alexander Graham Bell). In order to obtain more manageable numbers, we use the dB (decibel, where “deci” stands for one tenth) instead of the Bel for computation purposes. We have to multiply the Bel values by 10 (just as we need to multiply a distance by 1000 if we want to use millimeters instead of meters).

Example:

a = 33.01 dB, what is P1 / P2?

After first computing 33.01 / 10 = 3.301, we obtain:

## What does dBm mean?

If we refer an arbitrary power value to a fixed reference quantity, the logarithmic ratio of the two values yields a new absolute quantity. This quantity is defined as a level. The reference quantity most commonly used in telecommunications and radio frequency engineering is a power of 1 mW (one thousandth of one Watt) into 50 Ohm. The general power ratio P1 to P2 now becomes a ratio of P1 to 1 mW. The logarithmic ratio provides the level L. According to IEC 27 the reference value had to be indicated in the level index:

or the short form:

For example 5 mW corresponds to a level of LP/1mW = 6.99 dB.

To denote the reference of 1 mW, the ITU introduced the unit dBm. This unit is more common than the IEC 27 terminology, and will be used throughout this paper. With this, our example reads as follows:

To give you a feeling for the orders of magnitude which tend to occur, here are some examples: The output power range of signal generators extends typically from -140 dBm to +20 dBm or 0.01 fW (femto Watt) to 0.1 W. Mobile radio base stations transmit at +43 dBm or 20 W. Mobile phones transmit at +10 dBm to +33 dBm or 10 mW to 2 W. Broadcast transmitters operate at +70 dBm to +90 dBm or 10 kW to 1 MW.

## What is the difference between voltage decibels and power decibels?

First, please forget everything you have ever heard about voltage and power decibels. There is only one type of decibel, and it represents a ratio of two power levels P1 and P2. Of course, any power level can be expressed as a voltage if we know the resistance.

We can compute the logarithmic ratio as follows:

Using the following 3 familiar identities,

We obtain (again using lg to mean the base 10 logarithm):

Note the minus sign in front of the resistance term. In most cases, the reference resistance is equal for both power levels, i.e. R1 = R2.

Since

we can simplify as follows:

This also explains why we use 10·lg for power ratios and 20·lg for voltage ratios.

**Caution: (Very important!) This formula is valid only if R1 = R2. If, as sometimes occurs in television engineering, we need to take into account a conversion from 75 Ohm to 50 Ohm, we need to consider the ratio of the resistances. Conversion back to linear values is the same as before. For voltage ratios, we must divide the value a by 20 since we use U2 and decibels (20 = 2·10, 2 from U2, 10 from deci).**

## What is a level?

As we saw above, dBm involves a reference to a power level of 1 mW. Other frequently used reference quantities include 1 W, 1 V, 1 μV and also 1 A or 1 μA. According to IEC 27, they are designated as dB (W), dB (V), dB (μV), dB (A) and dB (μA), respectively, or in field strength measurements, dB (W/m2), dB (V/m), dB (μV/m), dB (A/m) and dB (μA/m). As was the case for dBm, the conventional way of writing these units according to ITU is dBW, dBV, dBμV, dBA, dBμA, dBW/m2, dBV/m, dBμV/m, dBA/m and dBμA/m. These units will be used in this paper. From the relative values for power level P1 (voltage U1) referred to power level P2 (voltage U2), we obtain absolute values using the reference values above. These absolute values are also known as levels. A level of 10 dBm means a value which is 10 dB above 1 mW, and a level of -17 dB(μV) means a value which is 17 dB below 1 μV. When computing these quantities, it is important to keep in mind whether they are power quantities or voltage quantities. Some examples of power quantities include power, energy, resistance, noise figure and power flux density.

Voltage quantities (also known as field quantities) include voltage, current, electric field strength, magnetic field strength and reflection coefficient.

Examples:

A power flux density of 5 W/m2 has the following level:

A voltage of 7 μV can also be expressed as a level in dB(μV):

Conversion from levels to linear values requires the following formulas:

## Attenuation and gain

The linear transfer function alin of a two-port circuit represents the ratio of the output power to the input power:

The transfer function is normally specified in dB:

If the output power P2 of a two-port circuit is greater than the input power P1, then the logarithmic ratio of P2 to P1 is positive. This is known as amplification or gain.

If the output power P2 of a two-port circuit is less than the input power P1, then the logarithmic ratio of P2 to P1 is negative. This is known as attenuation or loss (the minus sign is omitted).

Computation of the power ratio or the voltage ratio from the decibel value uses the following formulas:

Conventional amplifiers realize gains of up to 40 dB in a single stage, which corresponds to voltage ratios up to 100 and power ratios up to 10000. With higher values, there is a risk of oscillation in the amplifier. However, higher gain can be obtained by connecting multiple stages in series. The oscillation problem can be avoided through suitable shielding. The most common attenuators have values of 3 dB, 6 dB, 10 dB and 20 dB. This corresponds to voltage ratios of 0.7, 0.5, 0.3 and 0.1 or power ratios of 0.5, 0.25, 0.1 and 0.01. Here too, we must cascade multiple attenuators to obtain higher values. If we attempt to obtain higher attenuation in a single stage, there is a risk of crosstalk.

## Series connection of two-port circuits:

In the case of series connection (cascading) of two-port circuits, we can easily compute the total gain (or total attenuation) by adding the decibel values.

Cascading two-port circuits

The total gain is computed as follows:

Example:

The Fig. shows the input stages of a receiver. The total gain a is computed as follows: a = -0.7 dB + 12 dB – 7 dB + 23 dB = 27.3 dB.

# Conversion from decibels to percentage and vice versa

The term “percent” comes from the Latin and literally means “per hundred”. 1% means one hundredth of a value.

1 % of x = 0.01 x

When using percentages, we need to ask two questions:

▪ Are we calculating voltage quantities or power quantities?

▪ Are we interested in x% of a quantity or x% more or less of a quantity?

As mentioned above, voltage quantities are voltage, current, field strength and reflection coefficient, for example. Power quantities include power, resistance, noise figure and power flux density.

## Converting % voltage to decibels and vice versa

x% of a voltage quantity is converted to decibels as follows:

In other words: To obtain a value of x% in decibels, we must first convert the percentage value x to a rational number by dividing x by 100. To convert to decibels, we multiply the logarithm of this rational number by 20 (voltage quantity: 20) as shown above.

## Converting % power to decibels and vice versa

x% of a power quantity is converted to decibels as follows:

To obtain a value in decibels, we first convert the percentage value x to a rational number (as shown above) by dividing the number by 100. To convert to decibels (as described in section 2), we multiply the logarithm of this rational number by 10 (power quantity: 10).

# Using dB values in computations

This section demonstrates how to add power levels and voltages in logarithmic form, i.e. in decibels.

## Adding power levels

30 dBm + 30 dBm = 60 dBm? **Of course not!** If we convert these power levels to linear values, it is obvious that 1 W + 1 W = 2 W. This is 33 dBm and not 60 dBm. However, this is true only if the power levels to be added are uncorrelated. Uncorrelated means that the instantaneous values of the power levels do not have a fixed phase relationship with one another.

Note: Power levels in logarithmic units need to be converted prior to addition so that we can add linear values. If it is more practical to work with decibel values after the addition, we have to convert the sum back to dBm.

## Measuring signals at the noise limit

One common task involves measurement of weak signals close to the noise limit of a test instrument such as a receiver or a spectrum analyzer. The test instrument displays the sum total of the inherent noise and signal power, but it should ideally display only the signal power. The prerequisite for the following calculation is that the test instrument must display the RMS power of the signals. This is usually the case with power meters, but with spectrum analyzers, it is necessary to switch on the RMS detector.

First, we determine the inherent noise Lr of the test instrument by turning off the signal. Then, we measure the signal with noise Ltot. We can obtain the power P of the signal alone by subtracting the linear power values.

# Adding voltages

Likewise, we can add decibel values for voltage quantities only if we convert them from logarithmic units beforehand. We must also know if the voltages are correlated or uncorrelated. If the voltages are correlated, we must also know the phase relationship of the voltages.

Addition of two uncorrelated voltages

We add uncorrelated voltages quadratically, i.e. we actually add the associated power levels. Since the resistance to which the voltages are applied is the same for all of the signals, the resistance will disappear from the formula:

If the individual voltages are specified as levels, e.g. in dB (V), we must first convert them to linear values.

If the voltages are correlated, the computation becomes significantly more complicated. As we can see from the following figures, the phase angle of the voltages determines the total voltage, which is produced.

Addition of two correlated voltages, 0° phase angle

Blue represents voltage U1, green represents voltage U2 and red represents the total voltage U.

Addition of two correlated voltages, 90° phase angle

Addition of two correlated voltages, 180° phase angle

The total voltage U ranges from Umax = U1 + U2 for phase angle 0° (in-phase) to Umin = U1 – U2 for phase angle 180° (opposite phase). For phase angles in between, we must form the vector sum of the voltages (see elsewhere for more details).

Vector addition of two voltages

In actual practice, we normally only need to know the extreme values of the voltages, i.e. Umax and Umin.

If the voltages U1 and U2 are in the form of level values in dB (V) or dB (μV), we must first convert them to linear values just as we did with uncorrelated voltages. However, the addition is linear instead of quadratic (see the next section about peak voltages).

## Peak voltages

If we apply a composite signal consisting of different voltages to the input of an amplifier, receiver or spectrum analyzer, we need to know the peak voltage. If the peak voltage exceeds a certain value, limiting effects will occur which can result in undesired mixing products or poor adjacent channel power. The peak voltage U is equal to:

The maximum drive level for amplifiers and analyzers is usually indicated in dBm. In a 50 Ω system, conversion based on the peak voltage (in V) is possible with the following formula:

The factor 103 comes from the conversion from Watts to milli Watts

**Note that this power level represents the instantaneous peak power and not the RMS value of the power.**

# What do we measure in decibels?

This section summarizes some of the terms and measurement quantities, which are typically specified in decibels. This is not an exhaustive list and we suggest you consult the bibliography if you would like more information about this subject. The following sections are structured to be independent of one another so you can consult just the information you need.

## Signal-to-noise ratio (S/N)

One of the most important quantities when measuring signals is the signal-to-noise ratio (S/N). Measured values will fluctuate more if the S/N degrades. To determine the signal-to-noise ratio, we first measure the signal S and then the noise power N with the signal switched off or suppressed using a filter. Of course, it is not possible to measure the signal without any noise at all, meaning that we will obtain correct results only if we have a good S/N.

Sometimes, distortion is also present in addition to noise. In such cases, it is conventional to determine the signal to noise and distortion (SINAD) as opposed to just the signal-to-noise ratio.

Example:

We would like to measure the S/N ratio for an FM radio receiver. Our signal generator is modulated at 1 kHz with a suitable FM deviation. At the loudspeaker output of the receiver, we measure a power level of 100 mW, which represents both the signal and noise power. The noise power, which is measured next, must be subtracted from this quantity to determine the signal power. We now turn off the modulation on the signal generator and measure a noise power of 0.1 μW at the receiver output. The S/N is computed as follows:

To determine the SINAD value, we again modulate the signal generator at 1 kHz and measure (as before) a receiver power level of 100 mW. Now, we suppress the 1 kHz signal using a narrow notch filter in the test instrument. At the receiver output, all we now measure is the noise and the harmonic distortion. If the measured value is equal to, say, 0.5 μW, we obtain the SINAD as follows:

## Noise

Noise is caused by thermal agitation of electrons in electrical conductors. The power P which can be consumed by a sink (e.g. receiver input, amplifier input) is dependent on the temperature T and on the measurement bandwidth B (please do not confuse bandwidth B with B = Bel!).

Here, k is Boltzmann’s constant 1.38 x 10-23 JK-1 (Joules per Kelvin, 1 Joule = 1 Watt- Second), T is the temperature in K (Kelvin, 0 K corresponds to -273.15°C or – 459.67°F) and B is the measurement bandwidth in Hz. At room temperature (20°C, 68°F), we obtain per Hertz bandwidth a power of:

If we convert this power level to dBm, we obtain the following:

The thermal noise power at a receiver input is equal to -174 dBm per Hertz bandwidth. Note that this power level is not a function of the input impedance, i.e. it is the same for 50 Ω, 60 Ω and 75 Ω systems. The power level is proportional to bandwidth B. Using the bandwidth factor b in dB, we can compute the total power as follows:

Example:

An imaginary spectrum analyzer that produces no intrinsic noise is set to a bandwidth of 1 MHz. What noise power will it display?

The noise power level, which is displayed at room temperature at a 1 MHz bandwidth, is equal to -114 dBm. A receiver / spectrum analyzer produces 60 dB more noise with a 1 MHz bandwidth than with a 1 Hz bandwidth. A noise level of -114 dBm is displayed. If we want to measure lower amplitude signals, we need to reduce the bandwidth. However, this is possible only until we reach the bandwidth of the signal. To a certain extent, it is possible to measure signals even if they lie below the noise limit since each additional signal increases the total power, which is displayed (see the section on measuring signals at the noise limit above). However, we will quickly reach the resolution limit of the test instrument we are using. Certain special applications such as deep-space research and astronomy necessitate measurement of very low-amplitude signals from space probes and stars, for example. Here, the only possible solution involves cooling down the receiver input stages to levels close to absolute zero (-273.15°C or –459.67 F).

## Averaging noise signals

To display noise signals in a more stable fashion, it is conventional to switch on the averaging function provided in spectrum analyzers. Most spectrum analyzers evaluate signals using what is known as a sample detector and average the logarithmic values displayed on the screen. This results in a systematic measurement error since lower measured values have an excessive influence on the displayed measurement result. The following figure illustrates this effect using the example of a signal with sinusoidal amplitude modulation.

Amplitude-modulated signal with logarithmic amplitude values as a function of time

As we can see here, the sinewave is distorted to produce a sort of heart-shaped curve with an average value, which is too low, by 2.5 dB. R&S spectrum analyzers use an RMS detector to avoid this measurement error

## Noise factor, noise figure

The noise factor F of a two-port circuit is defined as the ratio of the input signal-tonoise ratio SNin to the output signal-to-noise ratio SNout.

The signal-to-noise ratio S/N is determined as described above. If the noise factor is specified in a logarithmic unit, we use the term noise figure (NF).

When determining the noise figure, which results from cascading two-port circuits, it is necessary to consider certain details, which are beyond the scope of this Application Note. Details can be found in the relevant technical literature or on the Internet

## Phase noise

An ideal oscillator has an infinitely narrow spectrum. Due to the different physical effects of noise, however, the phase angle of the signal varies slightly which results in a broadening of the spectrum. This is known as phase noise.

To measure this phase noise, we must determine the noise power of the oscillator PR as a function of the offset from the carrier frequency fc (known as the offset frequency fOffset) using a narrowband receiver or a spectrum analyzer in a bandwidth B. We then reduce the measurement bandwidth B computationally to 1 Hz. Now, we reference this power to the power of the carrier Pc to produce a result in dBc (1 Hz bandwidth). The c in dBc stands for “carrier”. We thus obtain the phase noise, or more precisely, the single sideband (SSB) phase noise L:

dBc is also a violation of the standard, but it is used everywhere. Conversion to linear power units is possible, but is not conventional. Data sheets for oscillators signal generators and spectrum analyzers typically contain a table with phase noise values at different offset frequencies. The values for the upper and lower sidebands are assumed equal.

Most data sheets contain curves for the single sideband phase noise ratio, which do not drop off so monotonically as the curve in the Fig. This is because the phase locked loops (PLLs) used in modern instruments to keep oscillators locked to a reference crystal oscillator result in an improvement but also a degradation of the phase noise as a function of the offset frequency due to certain design problems.

When comparing oscillators, it is also necessary to consider the value of the carrier frequency. If we multiply the frequency of an oscillator using a zero-noise multiplier (possible only in theory), the phase noise ratio will degrade proportionally to the voltage, i.e. if we multiply the frequency by 10, the phase noise will increase by 20 dB at the same offset frequency. Accordingly, microwave oscillators are always worse than RF oscillators as a general rule. When mixing two signals, the noise power levels of the two signals add up at each offset frequency.

## S parameters

Two-port circuits are characterized by four parameters: S11 (input reflection coefficient), S21 (forward transmission coefficient), S12 (reverse transmission coefficient) and S22 (output reflection coefficient).

S parameters for a two-port circuit

The S parameters can be computed from the wave quantities a1, b1 and a2, b2 as follows:

Wave quantities a and b are voltage quantities. If we have the S parameters in the form of decibel values, the following formulas apply:

## VSWR and reflection coefficient

Like the reflection coefficient, the voltage standing wave ratio (VSWR) or standing wave ratio (SWR) is a measure of how well a signal source or sink is matched to a reference impedance. VSWR has a range from 1 to infinity and is not specified in decibels. However, the reflection coefficient r is. The relationship between r and VSWR is as follows:

For VSWR = 1 (very good matching), r = 0. For a very high VSWR, r approaches 1 (mismatch or total reflection).

r represents the ratio of two voltage quantities. For r in decibels, we have ar:

ar is called return loss.

For computation of the VSWR from the reflection coefficient, r is inserted as a linear value. The following table shows the relationship between VSWR, r and ar/dB. If you just need a rough approximation of r from the VSWR, simply divide the decimal part of the VSWR in half. This works well for VSWR values up to 1.2.

Conversion from VSWR to reflection coefficient r and return loss ar

Note that for two-port circuits, r corresponds to the input reflection coefficient S11 or the output reflection coefficient S22.

Attenuators have the smallest reflection coefficients. Good attenuators have reflection coefficients <5% all the way up to 18 GHz. This corresponds to a return loss of > 26 dB or a VSWR < 1.1. Inputs to test instruments and outputs from signal sources generally have VSWR specifications <1.5, which corresponds to r < 0.2 or r > 14 dB.

## Field strength

For field strength measurements, we commonly see the terms power flux density, electric field strength and magnetic field strength. Power flux density S is measured in W/m2 or mW/m2. The corresponding logarithmic units are dB (W/m2) and dB (mW/m2).

Electric field strength E is measured in V/m or μV/m. The corresponding logarithmic units are dB (V/m) and dB (μV/m).

Conversion from dB (V/m) to dB (μV/m) requires the following formula:

Example:

-80 dB (V/m) = -80 dB(μV/m) + 120 dB = 40 dB(μV/m)

Magnetic field strength H is measured in A/m or μA/m. The corresponding logarithmic units are dB (A/m) and dB (μA/m).

Conversion from dB (A/m) to dB (μA/m) requires the following formula:

Example:

20 dB(μA/m) = 20 dB(A/m) – 120 dB = -100 dB(A/m)

## Antenna gain

Antennas generally direct electromagnetic radiation into a certain direction. The power gain G that results from this at the receiver is specified in decibels with respect to a reference antenna. The most common reference antennas are the isotropic radiator and the λ/2 dipole. The gain is specified in dBi or dBD. If the power gain is needed in linear units, the following formula can be used for conversion:

## Crest factor

The ratio of the peak power to the average thermal power (RMS value) of a signal is known as the crest factor. A sinusoidal signal has a peak value, which is 2 times greater than the RMS value, meaning the crest factor is 2, which equals 3 dB. For modulated RF signals, the crest factor is referred to the peak value of the modulation envelope instead of the peak value of the RF carrier signal. A frequencymodulated (FM) signal has a constant envelope and thus a crest factor of 1 (0 dB). If we add up many sinusoidal signals, the peak value can theoretically increase up to the sum of the individual voltages. The peak power Ps would then equal:

The RMS power P is obtained by adding up the individual power values:

We thus obtain a crest factor CF equal to:

The more (uncorrelated) signals we add up, the less probable it becomes that the total of the individual voltages will be reached due to the different phase angles. The crest factor fluctuates around a level of about 11 dB. The signal has a noise-like appearance.

A noise-like signal with a crest factor of 11 dB

Examples: The crest factor of noise is equal to approx. 11 dB. OFDM signals as are used in DAB, DVB-T and WLAN also have crest factors of approx. 11 dB. The CDMA signals stipulated by the CDMA2000 and UMTS mobile radio standards have crest factors ranging up to 15 dB, but they can be reduced to 7 dB to 9 dB using special techniques involving the modulation data. Except for bursts, GSM signals have a constant envelope due to the MSK modulation and thus a crest factor of 0 dB. EDGE signals have a crest factor of 3.2 dB due to the filter function of the 8PSK modulation

The following figure shows the so-called Complementary Cumulative Distribution Function (CCDF) of a noise like signal. The Crest Factor is that point of the measurement curve, where it reaches the x-axis. In the picture, this is at appr. 10.5 dB.

## Channel power and adjacent channel power

Modern communications systems such as GSM, CDMA2000 and UMTS manage a huge volume of calls. To avoid potential disruptions and the associated loss of revenue, it is important to make sure that exactly the permissible channel power Pch (where ch stands for channel) is available in the useful channel and no more. The power in the useful channel is most commonly indicated as the level Lch in dBm.

This is normally 20 W or 43 dBm.

In the adjacent channels, the power may not exceed the value Padj. This value ACPR (Adjacent Channel Power Ratio) is measured as a ratio to the power in the useful channel and is specified in dB.

Here, values of -40 dB (for mobile radio devices) down to -70 dB (for UMTS base stations) are required in the immediately adjacent channel and correspondingly higher values in the alternate channels. When measuring the power levels, it is important to consider the bandwidth of the channels. It can be different for the useful channel and the adjacent channel. Example (CDMA2000): Useful channel 1.2288 MHz, adjacent channel 30 kHz. Sometimes, it is also necessary to select a particular type of modulation filtering, e.g. square-rootcosine- roll-off. Modern spectrum analyzers have built-in measurement functions, which automatically take into account the bandwidth of the useful channel and adjacent channel as well as the filtering.

Adjacent channel power for a UMTS signal

## Modulation quality EVM

Ideally, we would like to be able to decode signals from digitally modulated transmitters with as few errors as possible in the receiver. Over the course of the transmission path, noise and interference are superimposed in an unavoidable process. This makes it all the more important for the signal from the transmitter to exhibit good quality. One measure of this quality is the deviation from the ideal constellation point. The figure below illustrates this based on the example of QPSK modulation.

To determine the modulation quality, the magnitude of the error vector Uerr is referenced to the nominal value of the modulation vector Umod. This quotient is known as the vector error or the error vector magnitude (EVM) and is specified as a percentage or in decibels.

We distinguish between the peak value EVMpeak occurring over a certain time interval and the RMS value of the error EVMRMS.

Note that these vectors are voltages. This means we must use 20·lg in our calculations. An EVM of 0.3% thus corresponds to -50 dB.

## Dynamic range of A/D and D/A converters

Important properties of analog to digital (A/D) and digital to analog (D/A) converters include the clock frequency fclock and the number of data bits n. For each bit, we can represent twice (or half, depending on our point of view) the voltage. We thus obtain a dynamic range D of 6 dB per bit (as we have already seen, 6 dB corresponds to a factor of 2 for a voltage quantity). There is also a system gain of 1.76 dB for measurement of sine shaped signals.

Example:

A 16-bit D/A converter has a dynamic range of 96.3 dB + 1.76 dB = 98 dB. In practice, A/D and D/A converters exhibit certain nonlinearities, which make it impossible to achieve their full theoretical values. In addition, clock jitter and dynamic effects mean that converters have a reduced dynamic range particularly at high clock frequencies. A converter is then specified using what is known as the spurious-free dynamic range or the number of effective bits.

Example:

An 8-bit A/D converter is specified as having 6.3 effective bits at a clock frequency of 1 GHz. It thus produces a dynamic range of 37.9 dB +1.76 dB = 40 dB. For a 1 GHz clock frequency, an A/D converter can handle signals up to 500 MHz (Nyquist frequency). If we use only a fraction of this bandwidth, we can actually gain dynamic range by using decimation filters. For example, an 8-bit converter can achieve 60 dB or more dynamic range instead of only 50 dB (= 8 · 6 + 1.76 dB). Based on the dynamic range, we can compute the number of effective bits as follows:

Example:

How many effective bits does an A/D converter have with a dynamic range of 70 dB? We compute as follows: 70 dB – 1.76 dB = 68.2 dB and 20log10(2) = 6.02 68.2 / 6.02 = 11.3 We thus obtain a result of 11.3 effective bits.

## dB (FS) (Full Scale)

Analog to digital converters and digital to analog converters have a maximum dynamic range, which is determined by the range of numbers they can process. For example, an 8-bit A/D converter can handle numbers from 0 to a maximum of 28 – 1= 255. This number is also known as the full-scale value (nFS). We can specify the drive level n of such converters with respect to this full-scale value and represent this ratio logarithmically.

If the converter is expected to represent positive and negative voltages, we must divide the range of values by two and take into account a suitable offset for the zero point.

# Few numbers worth knowing

Working with decibel values is a lot easier if you memorize a few key values. From just a few simple values, you can easily derive other values when needed. We can further simplify the problem by rounding exact values up or down to some easy to remember numbers. All we have to do is remember the simplified values, e.g. a power ratio of 2 corresponds to 3 dB (instead of the exact value of 3.02 dB which is rarely needed). The following table lists some of the most useful numbers to remember.

## Table for conversion between decibels and linear values

Conversion between decibels and linear values

**From this table, you should probably know at least the rough values for 3 dB, 6 dB, 10 dB and 20 dB by heart.**

Note: 3 dB is not an exact power ratio of 2 and 6 dB is not exactly 4! For everyday usage, however, these simplifications provide sufficient accuracy and as such are commonly used.

Intermediate values, which are not found in the table, can often be derived easily:

4 dB = 3 dB + 1 dB, corresponding to a factor of 2 + 20% of the power, i.e. approx. 2.4 times the power.

7 dB = 10 dB – 3 dB, corresponding to 10 times the power and then half, i.e. 5 times the power.

## Table for adding decibel values

If you need to compute the sum of two values specified in decibels precisely, you must convert them to linear form, add them and then convert them back to logarithmic form. However, the following table is useful for quick calculations. Column 1 specifies under Delta dB the difference between the two dB values. Column 2 specifies a correction factor for power quantities. Column 3 specifies a correction factor for voltage quantities. Add the correction factor to the higher of the two dB values to obtain the total.

Correction factors for adding decibel values

## Some more useful values

The following values are also useful under many circumstances:

**13 dBm**corresponds to URMS = 1 V into 50 Ω**0 dBm**corresponds to URMS = 0.224 V into 50 Ω**107 dB (μV)**corresponds to 0 dBm into 50 Ω**120 dB (μV)**corresponds to 1 V**-174 dBm**is the thermal noise power in 1 Hz bandwidth at a temperature of approx. 20 °C (68 °F).

# Smartphone Apps

### To ease calculations as the ones done in the preceding chapters, Rohde & Schwarz provides a range of apps for tablets and smartphones.

dB Calculator

The dB Calculator App comprises five independent calculation tools. dBm Calculator, Voltage Calculator, Unit Converter, dB Converter and VSWR Converter.

- Android: https://play.google.com/store/apps/details?id=com.rohdeschwarz.dbcalculator
- iOS: https://itunes.apple.com/app/id489100786
- Windows Phone: https://www.microsoft.com/store/apps/9nblggh0fwcd