Bandwidth (signal processing)

Bandwidthis the difference between the upper and lowerfrequenciesin a continuousband of frequencies.It is typically measured inunitofhertz(symbol Hz).

Amplitude (a) vs. frequency (f) graph illustratingbasebandbandwidth. Here the bandwidth equals the upper frequency.

It may refer more specifically to two subcategories:Passbandbandwidthis the difference between the upper and lowercutoff frequenciesof, for example, aband-pass filter,acommunication channel,or asignal spectrum.Basebandbandwidthis equal to the upper cutoff frequency of alow-pass filteror baseband signal, which includes a zero frequency.

Bandwidth in hertz is a central concept in many fields, includingelectronics,information theory,digital communications,radio communications,signal processing,andspectroscopyand is one of the determinants of the capacity of a givencommunication channel.

A key characteristic of bandwidth is that any band of a given width can carry the same amount ofinformation,regardless of where that band is located in thefrequency spectrum.^[a]For example, a 3 kHz band can carry a telephone conversation whether that band is at baseband (as in aPOTStelephone line) ormodulatedto some higher frequency. However, wide bandwidths are easier to obtain andprocessat higher frequencies because the§ Fractional bandwidthis smaller.

Overview

Bandwidth is a key concept in manytelecommunicationsapplications. Inradiocommunications, for example, bandwidth is the frequency range occupied by a modulatedcarrier signal.AnFM radioreceiver'stunerspans a limited range of frequencies. A government agency (such as theFederal Communications Commissionin the United States) may apportion the regionally available bandwidth tobroadcast licenseholders so that theirsignalsdo not mutually interfere. In this context, bandwidth is also known aschannel 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 offrequency response,degradation could, for example, mean more than 3dBbelow the maximum value or it could mean below a certain absolute value. As with any definition of thewidthof a function, many definitions are suitable for different purposes.

In the context of, for example, thesampling theoremandNyquist sampling rate,bandwidth typically refers tobasebandbandwidth. In the context ofNyquist symbol rateorShannon-Hartley channel capacityfor communication systems it refers topassbandbandwidth.

TheRayleigh bandwidthof 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]

Theessential bandwidthis defined as the portion of asignal spectrumin the frequency domain which contains most of the energy of the signal.^[2]

xdB bandwidth

The magnitude response of aband-pass filterillustrating the concept of −3 dB bandwidth at a gain of approximately 0.707

In some contexts, the signal bandwidth inhertzrefers to the frequency range in which the signal'sspectral 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 the3 dB point,that is the point where the spectral density is half its maximum value (or the spectral amplitude, in $\mathrm {V}$ or $\mathrm {V/{\sqrt {Hz}}}$ ,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 thesampling theorem.

The bandwidth is also used to denotesystem bandwidth,for example infilterorcommunication channelsystems. 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 anelectronic filteror 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 itscenter frequency,and in the low-pass filter is at or near itscutoff 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 samehalf-power gainconvention is also used inspectral width,and more generally for the extent of functions asfull width at half maximum(FWHM).

Inelectronic filterdesign, a filter specification may require that within the filterpassband,the gain is nominally 0 dB with a small variation, for example within the ±1 dB interval. In thestopband(s), the required attenuation in decibels is above a certain level, for example >100 dB. In atransition bandthe 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, thexdB point refers to the point where the gain isxdB below the nominal passband gain rather thanxdB below the maximum gain.

In signal processing andcontrol theorythe bandwidth is the frequency at which theclosed-loop system gaindrops 3 dB below peak.

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

The fact that in equivalentbasebandmodels 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 $B=2W$ ,where $B$ 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 $W$ is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require alow-pass filterwith cutoff frequency of at least $W$ to stay intact, and the physical passband channel would require a passband filter of at least $B$ to stay intact.

Relative bandwidth

The absolute bandwidth is not always the most appropriate or useful measure of bandwidth. For instance, in the field ofantennasthe 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( $B_{\mathrm {F} }$ ) andratio bandwidth( $B_{\mathrm {R} }$ ).^[4]In the following, the absolute bandwidth is defined as follows, $B=\Delta f=f_{\mathrm {H} }-f_{\mathrm {L} }$ where $f_{\mathrm {H} }$ and $f_{\mathrm {L} }$ 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 ( $f_{\mathrm {C} }$ ), $B_{\mathrm {F} }={\frac {\Delta f}{f_{\mathrm {C} }}}\,.$

The center frequency is usually defined as thearithmetic meanof the upper and lower frequencies so that, $f_{\mathrm {C} }={\frac {f_{\mathrm {H} }+f_{\mathrm {L} }}{2}}\$ and $B_{\mathrm {F} }={\frac {2(f_{\mathrm {H} }-f_{\mathrm {L} })}{f_{\mathrm {H} }+f_{\mathrm {L} }}}\,.$

However, the center frequency is sometimes defined as thegeometric meanof the upper and lower frequencies, $f_{\mathrm {C} }={\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}$ and $B_{\mathrm {F} }={\frac {f_{\mathrm {H} }-f_{\mathrm {L} }}{\sqrt {f_{\mathrm {H} }f_{\mathrm {L} }}}}\,.$

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]Fornarrowbandapplications, there is only marginal difference between the two definitions. The geometric mean version is inconsequentially larger. Forwidebandapplications 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, $\%B$ ), $\%B_{\mathrm {F} }=100{\frac {\Delta f}{f_{\mathrm {C} }}}\,.$

Ratio bandwidth

Ratio bandwidth is defined as the ratio of the upper and lower limits of the band, $B_{\mathrm {R} }={\frac {f_{\mathrm {H} }}{f_{\mathrm {L} }}}\,.$

Ratio bandwidth may be notated as $B_{\mathrm {R} }:1$ .The relationship between ratio bandwidth and fractional bandwidth is given by, $B_{\mathrm {F} }=2{\frac {B_{\mathrm {R} }-1}{B_{\mathrm {R} }+1}}$ and $B_{\mathrm {R} }={\frac {2+B_{\mathrm {F} }}{2-B_{\mathrm {F} }}}\,.$

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 inoctaves(i.e., as afrequency level) for wideband applications. An octave is a frequency ratio of 2:1 leading to this expression for the number of octaves, $\log _{2}\left(B_{\mathrm {R} }\right).$

Noise equivalent bandwidth

Setup for the measurement of the noise equivalent bandwidth

B_{n}

of the system with frequency response

H(f)

.

Thenoise equivalent bandwidth(orequivalent noise bandwidth (enbw)) of a system offrequency response $H(f)$ is the bandwidth of an ideal filter with rectangular frequency response centered on the system's central frequency that produces the same average power outgoing $H(f)$ when both systems are excited with awhite noisesource. The value of the noise equivalent bandwidth depends on the ideal filter reference gain used. Typically, this gain equals $|H(f)|$ at its center frequency,^[6]but it can also equal the peak value of $|H(f)|$ .

The noise equivalent bandwidth $B_{n}$ can be calculated in the frequency domain using $H(f)$ or in the time domain by exploiting theParseval's theoremwith the systemimpulse response $h(t)$ .If $H(f)$ is a lowpass system with zero central frequency and the filter reference gain is referred to this frequency, then:

$B_{n}={\frac {\int _{-\infty }^{\infty }|H(f)|^{2}df}{2|H(0)|^{2}}}={\frac {\int _{-\infty }^{\infty }|h(t)|^{2}dt}{2\left|\int _{-\infty }^{\infty }h(t)dt\right|^{2}}}\,.$

The same expression can be applied to bandpass systems by substituting theequivalent basebandfrequency response for $H(f)$ .

The noise equivalent bandwidth is widely used to simplify the analysis of telecommunication systems in the presence of noise.

Photonics

Inphotonics,the termbandwidthcarries 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 power meter) can operate
thedata rate(e.g., in Gbit/s) achieved in an optical communication system; seebandwidth (computing).

A related concept is thespectral linewidthof the radiation emitted by excited atoms.

Notes

^The information capacity of a channel depends onnoiselevel as well as bandwidth – seeShannon–Hartley theorem.Equal bandwidths can carry equal information only when subject to equalsignal-to-noise ratios.

References

^Jeffrey A. Nanzer,Microwave and Millimeter-wave Remote Sensing for Security Applications,pp. 268-269, Artech House, 2012ISBN 1608071723.
^Sundararajan, D. (4 March 2009).A Practical Approach to Signals and Systems.John Wiley & Sons. p. 109.ISBN 978-0-470-82354-5.
^ Van Valkenburg, M. E. (1974).Network Analysis(3rd ed.). Prentice-Hall. pp.383–384.ISBN 0-13-611095-9.Retrieved2008-06-22.
^Stutzman, Warren L.; Theiele, Gary A. (1998).Antenna Theory and Design(2nd ed.). New York.ISBN 0-471-02590-9.{{cite book}}:CS1 maint: location missing publisher (link)
^Hans G. Schantz,The Art and Science of Ultrawideband Antennas,p. 75, Artech House, 2015ISBN 1608079562
^Jeruchim, M. C.; Balaban, P.; Shanmugan, K. S. (2000).Simulation of Communication Systems. Modeling, Methodology, and Techniques(2nd ed.). Kluwer Academic.ISBN 0-306-46267-2.

[1] The information capacity of a channel depends onnoiselevel as well as bandwidth – seeShannon–Hartley theorem.Equal bandwidths can carry equal information only when subject to equalsignal-to-noise ratios.

[2] Jeffrey A. Nanzer,Microwave and Millimeter-wave Remote Sensing for Security Applications,pp. 268-269, Artech House, 2012ISBN 1608071723.

[3] Sundararajan, D. (4 March 2009).A Practical Approach to Signals and Systems.John Wiley & Sons. p. 109.ISBN 978-0-470-82354-5.

[4] Van Valkenburg, M. E. (1974).Network Analysis(3rd ed.). Prentice-Hall. pp.383–384.ISBN 0-13-611095-9.Retrieved2008-06-22.

[5] Stutzman, Warren L.; Theiele, Gary A. (1998).Antenna Theory and Design(2nd ed.). New York.ISBN 0-471-02590-9.{{cite book}}:CS1 maint: location missing publisher (link)

[6] Hans G. Schantz,The Art and Science of Ultrawideband Antennas,p. 75, Artech House, 2015ISBN 1608079562

[Jeruchim-7] Jeruchim, M. C.; Balaban, P.; Shanmugan, K. S. (2000).Simulation of Communication Systems. Modeling, Methodology, and Techniques(2nd ed.). Kluwer Academic.ISBN 0-306-46267-2.

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