# Bandwidth

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Bandwidth is a measure of frequency range and is typically measured in hertz. Bandwidth is a central concept in many fields, including information theory, radio communications, signal processing, and spectroscopy. Bandwidth is related to channel capacity for information transmission and is often confused with it. In particular, in common usage "bandwidth" also refers to data (information) transmission rates when communicating over certain media or devices.

##  Explanation

Bandwidth is a key concept in many 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.

There is no single universal precise definition of bandwidth, as it is vaguely understood to be a measure of how wide a function is in the frequency domain.

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). Another definition might not be so strict and ignore 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 it could mean below a certain absolute value. As with any definition of the width of a function, there are many definitions available, which are suitable for different applications.

According to the Shannon–Hartley theorem, the data rate of reliable communication is directly proportional to the frequency range of the signal used for the communication. In this context, the word bandwidth can refer to either the data rate or the frequency range of the communication system (or both).

##  Analog systems

Image:Bandwidth.png
A graph of a power spectral density, illustrating the concept of 3-dB (or half-power) bandwidth. The vertical axis here is proportional to power (square of Fourier magnitude); the frequency axis of this symbolic diagram can be linear or logarithmically scaled.

For analog signals, which can be mathematically viewed as functions of time, bandwidth $\Delta f$ 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 (half power $\approx$ 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. In the latter case, to say that a system has a certain bandwidth means that the system can process signals of that bandwidth.

A baseband bandwidth is 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 to −3 dB.

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 $\frac{1}{\sqrt{2}}$ of the maximum signal amplitude (half power).

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

##  Digital systems

In a digital communication system, bandwidth has a dual meaning. In the technical sense, it is slang for baud, the rate at which symbols may be transmitted through the system. It is also used in the colloquial sense to describe channel capacity, the rate at which bits may be transmitted through the system (see Shannon Limit). Hence, a 66 MHz digital data bus with 32 separate data lines may properly be said to have a bandwidth of 66 MHz and a capacity of 2.1 Gbit/s — but it would not be surprising to hear such a bus described as having a "bandwidth of 2.1 Gbit/s." Similar confusion exists for analog modems, where each symbol carries multiple bits of information so that a modem may transmit 56 kbit/s of information over a phone line with a bandwidth of only 12 kHz. A related metric which is used to measure the aggregated bandwidth of a whole network is bisection bandwidth.

In discrete time systems and digital signal processing, bandwidth is related to sampling rate according to the Nyquist-Shannon sampling theorem.

Bandwidth is also used in the sense of commodity, referring to something limited or something costing money. Thus, communication costs bandwidth, and improper use of someone else's bandwidth may be called bandwidth theft.

When Additive white Gaussian noise is present in a digital communication channel, the Shannon–Hartley theorem gives the relationship between the channel's bandwidth, the channel's capacity, and the Signal-to-noise ratio (SNR) ratio of the system.

##  Meaning of bandwidth in web hosting

In website hosting, bandwidth is the amount of information downloadable from the webserver over a prescribed period of time. In essence, it is the rate [data/time], but the time in this case is not seconds but rather a month or a week. Common connection rate terms such as 56K, broadband, etc. are also bandwidth, but are measured per second. Web hosting companies often quote a monthly bandwidth limit for a website, for example 100 GB/month. If visitors to the website download a total greater than 100 GB in one month, the bandwidth limit will have been exceeded.