X-dB Bandwidth
In some contexts, the signal bandwidth in hertz refers to the frequency range in which the signal's spectral density is nonzero or above a small threshold value. That definition is used in calculations of the lowest sampling rate that will satisfy the sampling theorem. Because this range of non-zero amplitude may be very broad or infinite, this definition is typically relaxed so that the bandwidth is defined as the range of frequencies in which the signal's spectral density is above a certain threshold relative to its maximum. Most commonly, bandwidth refers to the 3-dB bandwidth, that is, the frequency range within which the spectral density (in W/Hz or V2/Hz) is above half its maximum value (or the spectral amplitude, in V or V/Hz, is more than 70.7% of its maximum); that is, above −3 dB relative to the peak.
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, or that the system reduces the bandwidth of a white noise input to that bandwidth.
The 3 dB bandwidth of an electronic filter or communication channel is the part of the system's frequency response that lies within 3 dB of the response at its peak, which in the passband filter case is typically at or near its center frequency, and in the lowpass filter is near 0 hertz. If the maximum gain is 0 dB, the 3 dB gain is the range where the gain is more than -3dB, or the attenuation is less than + 3dB. This is also the range of frequencies where the amplitude gain is above 70.7% of the maximum amplitude gain, and above half the maximum power gain. This same "half power gain" convention is also used in spectral width, and more generally for extent of functions as full width at half maximum (FWHM).
In electronic filter design, a filter specification may require that within the filter passband, the gain is nominally 0 dB +/- a small number of dB, for example within the +/- 1 dB interval. In the stopband(s), the required attenuation in dB is above a certain level, for example >100 dB. In a transition band the gain is not specified. In this case, the filter bandwidth corresponds to the passband width, which in this example is the 1dB-bandwidth. If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.
A commonly used quantity is fractional bandwidth. This is the bandwidth of a device divided by its center frequency. E.g., a passband filter that has a bandwidth of 2 MHz with center frequency 10 MHz will have a fractional bandwidth of 2/10, or 20%.
In communication systems, in calculations of the Shannon–Hartley channel capacity, bandwidth refers to the 3dB-bandwidth. In calculations of the maximum symbol rate, the Nyquist sampling rate, and maximum bit rate according to the Hartley formula, the bandwidth refers to the frequency range within which the gain is non-zero, or the gain in dB is below a very large value.
The fact that in equivalent baseband models of communication systems, the signal spectrum consists of 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, where is the total bandwidth (i.e. the maximum passband bandwidth of the carrier-modulated RF signal and the minimum passband bandwidth of the physical passband channel), and is the positive bandwidth (the baseband bandwidth of the equivalent channel model). For instance, the baseband model of the signal would require a lowpass filter with cutoff frequency of at least to stay intact, and the physical passband channel would require a passband filter of at least to stay intact.
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 of the maximum signal amplitude (half power).
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