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Analog
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 wherein bandwidth would be the range of frequencies that the signal's Fourier transform has a power above a certain threshold, say 3 dB within the maximum value, in the frequency domain. Intuitively, 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.
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.
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 the signal is 1/Sqrt(2) of the maximum signal strength.
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Digital
For digital signals and 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 (in other words, 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 is usually 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.
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