Definitions

# Bandwidth (signal processing)

Bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a filter, a communication channel, or a signal spectrum, and is typically measured in hertz. In case of a baseband channel or signal, the bandwidth is equal to its upper cutoff frequency. Bandwidth in hertz is a central concept in many fields, including electronics, information theory, radio communications, signal processing, and spectroscopy.

In computer networking and other digital fields, the term bandwidth often refers to a data rate measured in bits per second, for example network throughput. The reason is that according to Hartley's law, the digital data rate limit (or channel capacity) of a physical communication link is related to its bandwidth in hertz, sometimes denoted analog bandwidth. For bandwidth as a computing term, less ambiguous terms are bit rate, throughput, goodput or channel capacity.

## Overview

Bandwidth is a key concept in many telephony applications. In radio communications, for example, bandwidth is the range of frequencies occupied by a modulated carrier wave, whereas in optics it is the width of an individual spectral line or the entire spectral range.

In many signal processing contexts, bandwidth is a valuable and limited resource. For example, 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 licensed broadcasters so that their signals do not mutually interfere. Each transmitter owns a slice of bandwidth, a valuable (if intangible) commodity.

For different applications there are different precise definitions. For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero. This would correspond to the mathematical notion of the support of a function (i.e., the total "length" of values for which the function is nonzero). A less strict and more practically useful definition will refer to the frequencies where the frequency function is small. Small could mean less than 3 dB below (i.e., less than half of) the maximum value, or more rarely 10 dB below, 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.

## Analog systems

For analog signals, which can be mathematically viewed as functions of time, bandwidth BW or $Delta f$ is the width, measured in hertz, of the frequency range in which the signal's Fourier transform is nonzero. Because this range of non-zero amplitude may be very broad, this definition is often relaxed so that the bandwidth is defined as the range of frequencies where the signal's Fourier transform has a power above a certain amplitude threshold, commonly half the maximum value, or −3 dB. Bandwidth of a signal is a measure of how rapidly its parameters (e.g., amplitude and phase) fluctuate with respect to time. Hence, the greater the bandwidth, the faster the variation in the signal parameters may be. The word bandwidth applies to signals as described above, but it could also apply to systems, for example filters or communication channels. To say that a system has a certain bandwidth means that the system can process signals of that bandwidth.

A baseband bandwidth is synonymous to the upper cutoff frequency, i.e. a specification of only the highest frequency limit of a signal. A non-baseband bandwidth is a difference between highest and lowest frequencies.

As an example, the (non-baseband) 3-dB bandwidth of the function depicted in the figure is $Delta f = f_2 - f_1 ,$, whereas other definitions of bandwidth would yield a different answer.

A commonly used quantity is fractional bandwidth. This is the bandwidth of a device divided by its center frequency. E.g., a device that has a bandwidth of 2 MHz with center frequency 10 MHz will have a fractional bandwidth of 2/10, or 20%.

The fact that real baseband systems have 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, and $W$ is the positive bandwidth. For instance, this signal would require a lowpass filter with cutoff frequency of at least $W$ to stay intact.

The 3 dB bandwidth of an electronic filter is the part of the filter's frequency response that lies within 3 dB of the response at its peak, which is typically at or near its center frequency.

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

In basic electric circuit theory when studying Band-pass and Band-reject filters the bandwidth represents the distance between the two points in the frequency domain where the signal is $frac\left\{1\right\}\left\{sqrt\left\{2\right\}\right\}$ of the maximum signal amplitude (half power).

## 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.