"If it can be written, or thought, it can be filmed..." STANLEY KUBRIC

# Bandwidth

Bandwidth is a measure of frequency range, measured in hertz, of a function of a frequency variable. Bandwidth is a central concept in many fields, including information theory, radio communications, signal processing, and spectroscopy. Bandwidth also refers to data 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

The frequency axis of this symbolic diagram would be logarithmically scaled.

For analog signals, which can be mathematically viewed as a function of time, bandwidth is the width, measured in hertz, of a frequency range in which the signal's Fourier transform is nonzero. This definition can be relaxed where the bandwidth would cover the range of frequencies where the signal's Fourier transform has a power above a certain amplitude threshold, say 3 dB within the maximum value, in the frequency domain. Bandwidth of a signal is a measure of how rapidly it fluctuates with respect to time. Hence, the greater the bandwidth, the faster the variation in the signal 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 is a short-hand for saying that the transfer function of the system has a certain bandwidth.

As an example, the 3 dB bandwidth of the function depicted in the figure is f2 - f1, 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 bandwidth of an electronic filter is the part of the filter's frequency response that lies within 3 dB of the response at the center frequency of its peak.

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 strength.

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.

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.

## 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. So this rate is not like 56K or broadband, etc., which are also bandwidth but are measured per second. Web hosting companies often quote a monthly bandwidth limit for a website, for example 2GB/month. If visitors to the website download a total greater than 2GB in one month, the bandwidth limit will have been exceeded.