# Bandwidth (signal processing)

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Bandwidth is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to passband bandwidth or baseband bandwidth. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

## Contents

Bandwidth in hertz is a central concept in many fields, including electronics, information theory, digital communications, radio communications, signal processing, and spectroscopy and is one of the determinants of the capacity of a given communication channel.

A key characteristic of bandwidth is that any band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum. [note 1] For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in a POTS telephone line) or modulated to some higher frequency.

## Overview

Bandwidth is a key concept in many telecommunications applications. In radio communications, for example, bandwidth is the frequency range occupied by a modulated carrier signal. An FM radio receiver's tuner spans a limited range of frequencies. A government agency (such as the Federal Communications Commission in the United States) may apportion the regionally available bandwidth to broadcast license holders so that their signals do not mutually interfere. In this context, bandwidth is also known as channel spacing.

For other applications, there are other definitions. One definition of bandwidth, for a system, could be the range of frequencies over which the system produces a specified level of performance. A less strict and more practically useful definition will refer to the frequencies beyond which performance is degraded. In the case of frequency response, degradation could, for example, mean more than 3  dB below the maximum value or it could mean below a certain absolute value. As with any definition of the width of a function, many definitions are suitable for different purposes.

In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth. In the context of Nyquist symbol rate or Shannon-Hartley channel capacity for communication systems it refers to passband bandwidth.

The Rayleigh bandwidth of a simple radar pulse is defined as the inverse of its duration. For example, a one-microsecond pulse has a Rayleigh bandwidth of one megahertz. [1]

The essential bandwidth is defined as the portion of a signal spectrum in the frequency domain which contains most of the energy of the signal. [2]

## x dB bandwidth

In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density (in W/Hz or V2/Hz) is nonzero or above a small threshold value. The threshold value is often defined relative to the maximum value, and is most commonly the , that is the point where the spectral density is half its maximum value (or the spectral amplitude, in ${\displaystyle \mathrm {V} }$ or ${\displaystyle \mathrm {V/{\sqrt {Hz}}} }$, is 70.7% of its maximum). [3] This figure, with a lower threshold value, can be used in calculations of the lowest sampling rate that will satisfy the sampling theorem.

The bandwidth is also used to denote system bandwidth, for example in filter or communication channel systems. To say that a system has a certain bandwidth means that the system can process signals with that range of frequencies, or that the system reduces the bandwidth of a white noise input to that bandwidth.

The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which, in the passband filter case, is typically at or near its center frequency, and in the low-pass filter is at or near its cutoff frequency. If the maximum gain is 0 dB, the 3 dB bandwidth is the frequency range where attenuation is less than 3 dB. 3 dB attenuation is also where power is half its maximum. This same half-power gain convention is also used in spectral width, and more generally for the extent of functions as full width at half maximum (FWHM).

In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In the stopband(s), the required attenuation in decibels is above a certain level, for example >100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1 dB-bandwidth. If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.

In signal processing and control theory the bandwidth is the frequency at which the closed-loop system gain drops 3 dB below peak.

In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3 dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley's law, the bandwidth refers to the frequency range within which the gain is non-zero.

The fact that in equivalent baseband models of communication systems, the signal spectrum consists of both negative and positive frequencies, can lead to confusion about bandwidth since they are sometimes referred to only by the positive half, and one will occasionally see expressions such as ${\displaystyle B=2W}$, where ${\displaystyle B}$ is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and ${\displaystyle W}$ is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require a low-pass filter with cutoff frequency of at least ${\displaystyle W}$ to stay intact, and the physical passband channel would require a passband filter of at least ${\displaystyle B}$ to stay intact.

## Relative bandwidth

The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field of antennas the difficulty of constructing an antenna to meet a specified absolute bandwidth is easier at a higher frequency than at a lower frequency. For this reason, bandwidth is often quoted relative to the frequency of operation which gives a better indication of the structure and sophistication needed for the circuit or device under consideration.

There are two different measures of relative bandwidth in common use: fractional bandwidth (${\displaystyle B_{\mathrm {F} }}$) and ratio bandwidth (${\displaystyle B_{\mathrm {R} }}$). [4] In the following, the absolute bandwidth is defined as follows,

${\displaystyle B=\Delta f=f_{\mathrm {H} }-f_{\mathrm {L} }}$

where ${\displaystyle f_{\mathrm {H} }}$ and ${\displaystyle f_{\mathrm {L} }}$ are the upper and lower frequency limits respectively of the band in question.

### Fractional bandwidth

Fractional bandwidth is defined as the absolute bandwidth divided by the center frequency (${\displaystyle f_{\mathrm {C} }}$),

${\displaystyle B_{\mathrm {F} }={\frac {\Delta f}{f_{\mathrm {C} }}}\ .}$

The center frequency is usually defined as the arithmetic mean of the upper and lower frequencies so that,

${\displaystyle f_{\mathrm {C} }={\frac {f_{\mathrm {H} }+f_{\mathrm {L} }}{2}}\ }$ and
${\displaystyle B_{\mathrm {F} }={\frac {2(f_{\mathrm {H} }-f_{\mathrm {L} })}{f_{\mathrm {H} }+f_{\mathrm {L} }}}\ .}$

However, the center frequency is sometimes defined as the geometric mean of the upper and lower frequencies,

${\displaystyle f_{\mathrm {C} }={\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}}$ and
${\displaystyle B_{\mathrm {F} }={\frac {f_{\mathrm {H} }-f_{\mathrm {L} }}{\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}}\ .}$

While the geometric mean is more rarely used than the arithmetic mean (and the latter can be assumed if not stated explicitly) the former is considered more mathematically rigorous. It more properly reflects the logarithmic relationship of fractional bandwidth with increasing frequency. [5] For narrowband applications, there is only marginal difference between the two definitions. The geometric mean version is inconsequentially larger. For wideband applications they diverge substantially with the arithmetic mean version approaching 2 in the limit and the geometric mean version approaching infinity.

Fractional bandwidth is sometimes expressed as a percentage of the center frequency (percent bandwidth, ${\displaystyle \%B}$),

${\displaystyle \%B_{\mathrm {F} }=100{\frac {\Delta f}{f_{\mathrm {C} }}}\ .}$

### Ratio bandwidth

Ratio bandwidth is defined as the ratio of the upper and lower limits of the band,

${\displaystyle B_{\mathrm {R} }={\frac {f_{\mathrm {H} }}{f_{\mathrm {L} }}}\ .}$

Ratio bandwidth may be notated as ${\displaystyle B_{\mathrm {R} }:1}$. The relationship between ratio bandwidth and fractional bandwidth is given by,

${\displaystyle B_{\mathrm {F} }=2{\frac {B_{\mathrm {R} }-1}{B_{\mathrm {R} }+1}}\ }$ and
${\displaystyle B_{\mathrm {R} }={\frac {2+B_{\mathrm {F} }}{2-B_{\mathrm {F} }}}\ .}$

Percent bandwidth is a less meaningful measure in wideband applications. A percent bandwidth of 100% corresponds to a ratio bandwidth of 3:1. All higher ratios up to infinity are compressed into the range 100–200%.

Ratio bandwidth is often expressed in octaves for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves,

${\displaystyle \log _{2}(B_{\mathrm {R} })\ .}$

## Photonics

In photonics, the term bandwidth occurs in a variety of meanings:

• the bandwidth of the output of some light source, e.g., an ASE source or a laser; the bandwidth of ultrashort optical pulses can be particularly large
• the width of the frequency range that can be transmitted by some element, e.g. an optical fiber
• the gain bandwidth of an optical amplifier
• the width of the range of some other phenomenon (e.g., a reflection, the phase matching of a nonlinear process, or some resonance)
• the maximum modulation frequency (or range of modulation frequencies) of an optical modulator
• the range of frequencies in which some measurement apparatus (e.g., a powermeter) can operate
• the data rate (e.g., in Gbit/s) achieved in an optical communication system; see bandwidth (computing).

A related concept is the spectral linewidth of the radiation emitted by excited atoms.

## Notes

1. Assuming equivalent noise level.

## Related Research Articles

Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The technology is used in telecommunications, radio broadcasting, signal processing, and computing.

In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission and a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital communication, used in applications such as digital television and audio broadcasting, DSL internet access, wireless networks, power line networks, and 4G/5G mobile communications.

In radio communications, single-sideband modulation (SSB) or single-sideband suppressed-carrier modulation (SSB-SC) is a type of modulation used to transmit information, such as an audio signal, by radio waves. A refinement of amplitude modulation, it uses transmitter power and bandwidth more efficiently. Amplitude modulation produces an output signal the bandwidth of which is twice the maximum frequency of the original baseband signal. Single-sideband modulation avoids this bandwidth increase, and the power wasted on a carrier, at the cost of increased device complexity and more difficult tuning at the receiver.

Baseband is a signal that has a near-zero frequency range, i.e. a spectral magnitude that is nonzero only for frequencies in the vicinity of the origin and negligible elsewhere. In telecommunications and signal processing, baseband signals are transmitted without modulation, that is, without any shift in the range of frequencies of the signal. Baseband has a low frequency—contained within the band from close to zero hertz up to a higher cut-off frequency. Baseband can be synonymous with lowpass or non-modulated, and is differentiated from passband, bandpass, carrier-modulated, intermediate frequency, or radio frequency (RF).

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.

A passband is the range of frequencies or wavelengths that can pass through a filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all the radio waves picked up by its antenna. The passband of a receiver is the range of frequencies it can receive when it is tuned into the desired frequency (channel).

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in decibels. A ratio higher than 1:1 indicates more signal than noise.

The total harmonic distortion is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency. Distortion factor, a closely related term, is sometimes used as a synonym.

In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy-channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free information per time unit that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density. The law is named after Claude Shannon and Ralph Hartley.

In telecommunications, frequency-division multiplexing (FDM) is a technique by which the total bandwidth available in a communication medium is divided into a series of non-overlapping frequency bands, each of which is used to carry a separate signal. This allows a single transmission medium such as a cable or optical fiber to be shared by multiple independent signals. Another use is to carry separate serial bits or segments of a higher rate signal in parallel.

In physics and engineering, the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is approximately defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results. Higher Q indicates a lower rate of energy loss and the oscillations die out more slowly. A pendulum suspended from a high-quality bearing, oscillating in air, has a high Q, while a pendulum immersed in oil has a low one. Resonators with high quality factors have low damping, so that they ring or vibrate longer.

In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate, but is still able to reconstruct the signal.

Chebyshev filters are analog or digital filters having a steeper roll-off than Butterworth filters, and have passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters".

In signal processing, a matched filter is obtained by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise.

In digital communication or data transmission, Eb/N0 is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing the bit error rate (BER) performance of different digital modulation schemes without taking bandwidth into account.

Pulse compression is a signal processing technique commonly used by radar, sonar and echography to increase the range resolution as well as the signal to noise ratio. This is achieved by modulating the transmitted pulse and then correlating the received signal with the transmitted pulse.

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

Staggered tuning is a technique used in the design of multi-stage tuned amplifiers whereby each stage is tuned to a slightly different frequency. In comparison to synchronous tuning it produces a wider bandwidth at the expense of reduced gain. It also produces a sharper transition from the passband to the stopband. Both staggered tuning and synchronous tuning circuits are easier to tune and manufacture than many other filter types.

A double-tuned amplifier is a tuned amplifier with transformer coupling between the amplifier stages in which the inductances of both the primary and secondary windings are tuned separately with a capacitor across each. The scheme results in a wider bandwidth and steeper skirts than a single tuned circuit would achieve.

Carrier frequency offset (CFO) is one of many non-ideal conditions that may affect in baseband receiver design. In designing a baseband receiver, we should notice not only the degradation invoked by non-ideal channel and noise, we should also regard RF and analog parts as the main consideration. Those non-idealities include sampling clock offset, IQ imbalance, power amplifier, phase noise and carrier frequency offset nonlinearity.

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