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# Bandwidth

 Contents

## Analog

Bandwidth is the width, usually measured in hertz, of a frequency band f2 − f1. It can also be used to describe a signal, in which case the meaning is the width of the smallest frequency band within which the signal can fit.

It is usually notated B, W, or BW. 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 compared to 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.

## Digital

By extension from the above, the word bandwidth is also used to mean the amount of data that can be transferred through a digital connection in a given time period (i.e., the connection's bit rate). In such cases, bandwidth is usually measured in bits or bytes per second.

In the physical world, a digital signal must be represented in an analog form for actual transmission. This can be a complex process. First the bit pattern must undergo a suitable form of channel coding, appropriate to the expected noise level of the analog channel. Then it must be transformed into an analog waveform using line coding, and modulated onto a carrier signal. The latter two processes depend upon the actual nature of the transmission medium, whether it be electrical, optical or electromagnetic.

Mathematically, the maximum digital bit rate for a given analog bandwidth and noise level is determined by the Shannon-Hartley theorem. How closely this is approximated depends to a great extent upon the choice of channel coding, which must introduce just enough redundancy to match the noise level. Too little redundancy, and expensive retransmissions will reduce the useful bitrate. Too much, and the error-correction overhead will reduce the bitrate left over for the signal. The Shannon-Hartley limit is approached closely by Reed-Solomon codes used on optical media, and even more closely by Turbo codes used in satellite communication.

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