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dBc

dBc, or decibels relative to the carrier, is a unit of measurement in radio frequency (RF) engineering that expresses the power level of a signal component as a logarithmic ratio relative to the power of the carrier signal, typically denoted in decibels (dB).^[1] This relative measure is essential for quantifying disturbances or secondary signals in modulated systems, where a value of 0 dBc indicates equal power to the carrier, positive values signify greater power, and negative values—such as -30 dBc—denote power 1,000 times lower than the carrier.^[2] Commonly applied in assessing signal quality, dBc is used to evaluate phase noise, which represents random fluctuations in the phase of a signal and is often specified as dBc/Hz to normalize for bandwidth, for example, -80 dBc/Hz at a 10 kHz offset from the carrier in oscillator performance metrics.^[3] It also measures harmonics and spurious signals, such as a second harmonic at -50 dBc, indicating its power is 50 dB below the carrier, helping engineers ensure compliance with spectral purity standards in transmitters and amplifiers.^[3] Unlike absolute power units like dBm (decibels relative to 1 milliwatt), dBc provides a carrier-referenced comparison without needing an external power reference, making it ideal for relative assessments in complex RF environments.^[2]

Fundamentals

Definition

dBc, or decibels relative to the carrier, is a unit that quantifies the power of a signal component—such as noise, a sideband, or a spur—relative to the power of the main carrier signal.^[2]^[1] This relative measure expresses the ratio in decibels, where the carrier serves as the reference power level. The carrier signal itself is the primary, unmodulated tone or the dominant component in a modulated waveform, providing a consistent benchmark for comparison.^[4] It finds primary application in radiofrequency (RF) engineering, telecommunications, and analog/digital signal processing, enabling the description of signal quality and interference levels without needing absolute power references.^[5]^[1] In this context, dBc allows engineers to assess how much weaker or stronger unwanted components are compared to the desired carrier, facilitating standardized evaluations across systems.^[4] Values in dBc can be positive or negative: positive indicates the component's power exceeds the carrier's, while negative values—prevalent in practical scenarios—denote it is weaker, such as -30 dBc meaning the component's power is one-thousandth that of the carrier.^[2]^[1] The decibel scale underlying dBc represents a logarithmic ratio of powers.^[2]

Notation and Units

The standard notation for the basic power ratio in dBc expresses the level of a signal component relative to the carrier power, using the lowercase "c" to denote the carrier as the reference. This convention distinguishes dBc from other relative decibel units, such as dBFS (decibels relative to full scale, used in digital signal processing for maximum signal amplitude) or dBi (decibels relative to an isotropic radiator, applied in antenna gain measurements).^[6]^[5] In standards from organizations like IEEE and ITU, dBc is employed to quantify sideband suppression and harmonic levels relative to the carrier, ensuring consistent reporting of spectral purity in radio frequency systems. For instance, IEEE Std 1139-2022 formalizes dBc for expressing deviations in frequency and phase stability measurements. Similarly, ITU-R recommendations use dBc to assess unwanted emissions and distortion in communication systems.^[7]^[8] When normalized to bandwidth, particularly for spectral density in phase noise analysis, the notation extends to dBc/Hz, indicating the power per hertz relative to the carrier. This normalization divides the noise power by the bandwidth in hertz, facilitating comparisons of noise floors across different measurement conditions and frequency offsets. IEEE Std 1139-2022 defines this as the single-sideband phase noise power spectral density, denoted as L(f) in dBc/Hz, where f is the offset frequency. The ITU Handbook on precise frequency systems similarly describes dBc/Hz as the phase-noise power in a 1 Hz bandwidth relative to the carrier, standard for oscillator characterization.^[7]^[9]

Mathematical Basis

Calculation Formula

The dBc (decibels relative to the carrier) quantifies the power ratio of a signal component, such as a spurious emission or noise, relative to the carrier signal power using the logarithmic scale defined by:

\text{dBc} = 10 \log_{10} \left( \frac{P_{\text{component}}}{P_{\text{carrier}}} \right)

where P_{\text{component}} represents the power of the specific signal component (e.g., a harmonic or phase noise contribution) and P_{\text{carrier}} denotes the power of the unmodulated carrier signal, with both powers measured in the same linear units such as watts or milliwatts.^[1] This expression ensures that the result is always negative or zero for components weaker than or equal to the carrier, reflecting the relative suppression.^[6] The formula derives directly from the general decibel definition for power ratios, \text{dB} = 10 \log_{10} \left( \frac{P_1}{P_2} \right), where the reference power P_2 is specialized to the carrier power P_{\text{carrier}} instead of a fixed value like 1 milliwatt (as in dBm).^[2] This specialization maintains the logarithmic compression of wide dynamic ranges while anchoring measurements to the signal's own carrier strength, which varies with the system under test.^[5] To apply the formula, powers must first be converted to a linear scale if initially measured in decibels; for instance, if voltages are available, power proportionality to the square of voltage requires adjusting to $20 \log_{10} \left( \frac{V_{\text{component}}}{V_{\text{carrier}}} \right) for equivalent power ratios, assuming identical impedances.^[2] In edge cases, the formula yields interpretable limits: when P_{\text{component}} = 0, the argument of the logarithm is zero, so \log_{10}(0) \to -\infty and thus dBc approaches -\infty, indicating complete absence of the component relative to the carrier.^[5] Conversely, if P_{\text{component}} = P_{\text{carrier}}, the ratio equals 1, \log_{10}(1) = 0, and dBc = 0, signifying equal power levels.^[6] These behaviors stem inherently from the properties of the base-10 logarithm applied to power ratios greater than or equal to zero.^[2]

Relation to Power Ratios

The dBc scale, being a logarithmic measure of power relative to the carrier, directly corresponds to linear power ratios through the inverse transformation, where the linear power ratio r is given by r = 10^{{\rm dBc}/10}.^[10] For instance, a dBc value of -20 indicates a linear power ratio of 0.01, meaning the signal component carries only 1% of the carrier's power.^[11] This conversion is essential in signal analysis to quantify the relative strength of unwanted components, such as harmonics or intermodulation products, in terms of their fractional contribution to the total power.^[11] In practical RF systems, dBc thresholds provide benchmarks for signal integrity; for example, spurious emissions in transmitters are often required to remain below -60 dBc to minimize interference and maintain acceptable signal-to-noise ratios (SNR), as levels exceeding this can degrade receiver performance by introducing noise-like distortions equivalent to a 60 dB penalty in SNR under certain conditions. International standards, such as those from the ITU, specify limits around -70 dBc for many applications to ensure spurs do not exceed a small fraction of carrier power, thereby preserving overall system dynamic range. The logarithmic nature of dBc excels at compressing vast dynamic ranges into manageable scales; a linear power ratio of $10^6 (one million times weaker than the carrier) translates to just -60 dBc, allowing engineers to visualize and compare signals spanning orders of magnitude without numerical overflow in analysis tools.^[10] This property is particularly valuable in RF engineering, where carrier powers might range from milliwatts to watts while spurs could be in the nanowatt regime, enabling concise representation of system performance.^[11] However, dBc measurements assume coherent signal detection, where the carrier and relative components are phase-aligned during acquisition, which can lead to inaccuracies in non-coherent or pulsed scenarios.^[12] Additionally, in broadband contexts—such as wideband noise or emissions—dBc values may not accurately reflect peak amplitudes if the spectrum analyzer's resolution bandwidth integrates over the signal, unlike narrowband measurements where discrete tones are resolved sharply, potentially underestimating interference in EMI assessments.^[13]

Applications

Phase Noise Measurement

Phase noise refers to the random fluctuations in the instantaneous phase of a sinusoidal signal, typically arising from thermal noise, flicker noise, or other instabilities in oscillators and transmitters. These fluctuations are quantified in the frequency domain as the power spectral density of the phase noise relative to the carrier signal power, providing a measure of signal purity essential for high-frequency applications.^[14] The standard metric for phase noise is denoted as L(f), defined as one-half the power spectral density of the phase fluctuations S_\phi(f), normalized to the carrier power and expressed in units of dBc/Hz at an offset frequency f from the carrier. This represents the single-sideband noise power in a 1 Hz bandwidth, relative to the carrier power on a logarithmic scale. For instance, a specification of -100 dBc/Hz at a 10 kHz offset means the noise power density is 100 dB below the carrier level per Hertz of bandwidth, a common benchmark for evaluating oscillator performance in microwave systems.^[14]^[15] Phase noise measurements are commonly performed using two primary setups: the direct spectrum analyzer method and the phase detector method. In the direct approach, the device under test is fed into a spectrum analyzer tuned to the carrier frequency, where the analyzer's resolution bandwidth filter captures the noise sidebands, requiring corrections for instrument noise floor and video filtering effects to achieve accurate readings. The phase detector method, more sensitive for low-noise signals, involves mixing the test signal with a low-noise reference oscillator in quadrature, followed by baseband spectrum analysis to isolate phase fluctuations; this setup often integrates over a defined bandwidth to compute total phase jitter. These techniques enable precise characterization, with integration yielding metrics like root-mean-square phase error for overall noise assessment.^[16]^[17] The measurement of phase noise in dBc/Hz is crucial for ensuring clock stability in communication systems, such as GPS receivers where excessive phase noise can degrade positioning accuracy, and 5G networks where it limits MIMO performance and increases error rates in high-data-rate links. Additionally, phase noise spectra relate directly to time-domain stability measures like Allan variance, which quantifies frequency deviations over averaging intervals and helps identify dominant noise processes for long-term oscillator reliability.^[18]^[19]^[20]

Spurious-Free Dynamic Range

Spurious-free dynamic range (SFDR) quantifies the ratio of the power of the fundamental carrier signal to the power of the strongest spurious signal in the output spectrum, typically measured in dBc to indicate the level of the largest spur relative to the carrier. In digital systems, such as those involving analog-to-digital converters (ADCs), SFDR may alternatively be expressed in dBFS, referencing the full-scale signal level. This metric assesses the purity of the signal by identifying the dynamic range over which spurious components remain below a detectable threshold, ensuring minimal interference from unwanted distortions. In ADCs and digital-to-analog converters (DACs), SFDR is determined by the maximum dBc value among all spurious signals, which is often limited by harmonic distortions or intermodulation products generated during signal processing. For instance, in high-speed data converters, third-order intermodulation products from nonlinearities can dominate, reducing the effective SFDR and impacting overall system performance. SFDR plays a critical role in RF mixers and data converters, where maintaining high values—such as 70 dBc—ensures that the output remains clean, with the largest spurious signal suppressed by at least 70 dB relative to the carrier, thereby supporting reliable signal transmission in communication systems. Common testing methods for SFDR include two-tone tests, in which two sinusoids of equal amplitude and closely spaced frequencies are applied to the input to generate and measure intermodulation distortion products. These tests reveal how nonlinearities produce spurs that degrade SFDR, particularly in bandwidth-limited scenarios. In ADCs, SFDR correlates with bit resolution, providing approximately 6 dB of dynamic range per effective bit, which helps estimate the converter's ability to resolve signals without spurious interference dominating weaker components.

Comparisons

dBc Versus Absolute Decibel Units

dBm represents an absolute unit of power measurement, defined as decibels relative to 1 milliwatt, where 0 dBm corresponds exactly to 1 mW of power.^[21] This scale allows for the quantification of signal strength in absolute terms, facilitating comparisons against fixed references like device specifications or environmental noise floors.^[2] In contrast, dBc is a relative unit that measures the power of a signal component, such as a harmonic or spur, as a ratio to the carrier power, expressed in decibels.^[22] The key difference lies in their referencing: dBc provides a carrier-independent ratio in value but remains context-dependent on the system's carrier level, whereas dBm demands calibration to an absolute power scale for meaningful interpretation.^[2] This makes dBc ideal for assessing relative performance without needing precise absolute power knowledge. The conversion between dBc and dBm is straightforward, given by the formula

\mathrm{dBc}_\mathrm{component} = \mathrm{dBm}_\mathrm{component} - \mathrm{dBm}_\mathrm{[carrier](/page/Carrier)}

where the result indicates how many decibels the component is below the carrier.^[22] For example, if the carrier is at 10 dBm and a spur measures -20 dBm, the spur level is -30 dBc.^[22] dBc is typically used for system-internal evaluations, such as determining modulation depth or the relative amplitude of sidebands in modulated signals.^[2] Conversely, dBm is essential for applications requiring absolute quantification, including regulatory compliance where emission limits are specified in terms like -13 dBm/MHz outside authorized bandwidths.^[23]

dBc Versus Other Relative Units

dBFS (decibels relative to full scale) is a relative unit commonly employed in digital signal processing, particularly within analog-to-digital converters (ADCs) and digital audio systems, where it quantifies signal levels with respect to the maximum representable amplitude before clipping occurs.^[24] In contrast to dBc, which benchmarks power against a specific carrier signal in RF contexts, dBFS uses the system's full-scale capacity as the fixed reference, enabling assessments of headroom and dynamic range independent of any particular signal component.^[25] This distinction is critical in applications like ADC performance evaluation, where spurious-free dynamic range (SFDR) may be specified in either dBFS (relative to the ADC's input range) or dBc (relative to the test carrier tone), with conversions required for direct comparisons—such as subtracting the carrier's offset from full scale to align scales.^[26] dBi (decibels relative to isotropic) and dBd (decibels relative to dipole) are relative units specific to antenna performance, measuring directional gain compared to theoretical reference radiators: an isotropic point source for dBi and a half-wave dipole for dBd, with dBi values typically 2.15 dB higher than equivalent dBd figures.^[3] Unlike dBc, which expresses the ratio of a signal's power to that of a carrier within the same transmission, dBi and dBd focus on spatial radiation patterns and efficiency, not intra-system power comparisons.^[27] These units are thus suited to antenna design and link budget calculations in RF systems, where gain in a particular direction informs coverage, but they do not directly relate to noise or distortion relative to a modulated carrier as dBc does. The choice between dBc and other relative units like dBFS or dBi depends on the measurement context: dBc is preferred for analyzing phase noise, spurs, or intermodulation in modulated carrier signals, such as those in AM or FM transmitters, where the carrier provides a natural reference for relative purity.^[28] Conversely, dBFS is selected in digital domains to monitor signal levels against clipping thresholds, ensuring operational margins in ADCs or audio processing chains.^[29] Antenna-related units like dBi apply when evaluating propagation efficiency rather than signal integrity. A common pitfall arises from applying dBc in scenarios lacking a well-defined carrier, such as unmodulated continuous-wave signals or multi-tone tests without a primary reference, which can result in ambiguous or inconsistent specifications across systems.^[30] This misuse contrasts with the clearer references in dBFS (system maximum) or dBi (standard radiator), highlighting the need for explicit carrier identification in dBc-based metrics to maintain comparability.

References

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Apr 16, 2018 · dBc is the ratio of input signal power vs carrier signal power, used to measure phase noise and harmonics relative to carrier power.
[2]
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What it is: dBc, or decibels relative to the carrier, is used to express the power of a signal relative to the power of a carrier signal. Think of it as ...
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This switch can control 1 KW transmitter power with excellent distortion performance ( IM3 < -80 dBc). ... [1] IEEE Standard Dictionary of Electrical and ...
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This difference in power is usually given in dBc, i.e., the number of decibels down from the transmit or “carrier” channel. ACLR quantifies ...
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The dBc unit expresses power with respect to the power of a related signal. The dBi unit expresses the gain of an antenna relative to the response of an ...
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Dec 8, 2022 · This standard defines and formalizes the techniques and physical quantities used to measure and characterize frequency, phase, and amplitude ...
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) The units of both are dBc/Hz; that is, the amount of phase-noise power in a bandwidth of one Hertz, relative to the power in the carrier. Phase noise on a ...
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dBc. dB relative to the carrier power. dBc/Hz. dB relative to the carrier power per Hertz of bandwidth. dBm. Absolute unit of measure in decibels: 0dBm = 1 mW.
[22]
None
### Summary of dBc and dBm from http://www.ittc.ku.edu/~jstiles/622/handouts/Harmonics%20spurs%20and%20dBc.pdf
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Compliance with these rules is based on the use of measurement ... emission outside the licensee's authorized bandwidth shall not exceed −13 dBm/MHz.
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Jul 14, 2014 · Because the test signal (carrier) is 1dB below Full-Scale (-1dBFS), you would subtract 1dB from SFDR(dBFS) to get SFDR(dBc). For Ain = -1dBFS, ...<|control11|><|separator|>
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