The bandwidth frequency calculator above describes a band in the four ways it usually gets quoted. Give it a lower and an upper cutoff and it returns the bandwidth, the geometric centre frequency, the quality factor and the fractional bandwidth, plus the span in octaves. Give it a centre frequency and a Q and it returns the cutoffs that band would have.
Arb Digital has several tools with bandwidth in the name and they do quite different things. The bandwidth converter rescales data rates between bits and bytes per second; the bandwidth requirement calculator sizes a network connection for a workload; the bandwidth delay product calculator handles the window sizing problem on a long network path. This page is the signal-processing one: bandwidth as a span of frequencies, in hertz, defined by half-power points.
What This Bandwidth Frequency Calculator Does
Bandwidth in this sense is the width of the frequency range a system passes. It is defined by the two frequencies at which the output power has fallen to half its peak, which in decibels is a drop of 3.01 dB, and which in voltage terms is a fall to about 70.7 per cent. Those are the cutoff or corner frequencies, and everything else on this page is derived from them.
The quality factor Q is the centre frequency divided by the bandwidth. It is a measure of selectivity: a high Q means a narrow band relative to where it sits, and a sharp peak. Fractional bandwidth is the reciprocal of Q expressed as a percentage, and octave span expresses the same width on a logarithmic scale, which is the natural one for audio and for anything spanning more than an octave or two.
The tool also reports the damping ratio, which is what control engineers use where circuit engineers use Q, in the note under the grid. The two describe the same thing: the damping ratio is one divided by twice Q, so a Q of 10 is a damping ratio of 0.05, and a Q of 0.5 is critical damping.
How to Use It
- Check that your cutoffs really are half-power points. Some datasheets quote a passband at 1 dB or 0.5 dB ripple instead, and those give a narrower band and a higher apparent Q.
- Put both frequencies in the same unit. The unit selector is presentational, and Q and fractional bandwidth come out dimensionless either way.
- Read the geometric centre rather than assuming the midpoint. For a wide band the two differ substantially, and Q is defined against the geometric one.
- Switch to the second mode to design rather than describe. Enter the centre frequency you want and the Q you need, and the cutoffs come back.
- Use the octave span when the band is wide. A band from 20 Hz to 20 kHz has a Q below one, which is a nearly useless number; ten octaves is a much better description of it.
The Formula: How Bandwidth and Q Are Calculated
Bandwidth is simply BW = f2 − f1. The centre frequency for a resonant band is the geometric mean, f0 = √(f1f2), and the quality factor is Q = f0 ÷ BW. Fractional bandwidth is BW ÷ f0, the reciprocal of Q, and the octave span is log2(f2 ÷ f1).
OpenStax University Physics Volume 2, section 15.5 on resonance in an AC circuit, defines bandwidth as the range of angular frequencies over which the average power exceeds half its maximum, and gives the quality factor as the resonant frequency divided by that bandwidth, noting that a high Q means a sharp resonance peak. The HyperPhysics page on the RLC series circuit covers the circuit that produces such a band, where impedance is minimised and the phase angle is zero at resonance.
Going the other way, from a centre frequency and Q to the cutoffs, the exact relations are f1 = f0(√(1 + 1 ÷ 4Q²) − 1 ÷ 2Q) and f2 = f0(√(1 + 1 ÷ 4Q²) + 1 ÷ 2Q). These are not the approximation f0 ± BW ÷ 2, which is only close for high Q.
Work the defaults. With cutoffs at 950 and 1,050 Hz, the bandwidth is 100 Hz. The geometric centre is √(950 × 1,050) = √997,500 = 998.75 Hz, against an arithmetic midpoint of 1,000 Hz. Q is 998.75 ÷ 100 = 9.9875, the fractional bandwidth is 10.01 per cent, and the span is log2(1,050 ÷ 950) = 0.1444 octaves. Running it backwards from f0 = 1,000 Hz and Q = 10 gives √(1 + 0.0025) = 1.00125, so the cutoffs come out at 951.25 and 1,051.25 Hz — exactly 100 Hz apart, with a geometric mean back at 1,000 Hz.
Why the Centre Is Geometric and Not Arithmetic
This is the single most common error in bandwidth work, and it is invisible at narrow bandwidths, which is what makes it dangerous.
A resonant system is symmetric on a logarithmic frequency axis, not a linear one. The response at half an octave below resonance mirrors the response at half an octave above it. Half an octave below 1,000 Hz is 707 Hz and half an octave above is 1,414 Hz, and the midpoint of those two on a linear scale is 1,061 Hz, which is not the resonant frequency at all. The geometric mean of 707 and 1,414 is 1,000, which is.
For the defaults on this page the geometric centre is 998.75 Hz against an arithmetic 1,000 Hz, a difference of about a tenth of a per cent that nobody would notice. For a band from 100 Hz to 10,000 Hz the geometric centre is 1,000 Hz and the arithmetic midpoint is 5,050 Hz, and using the wrong one gives a Q five times too large. The rule to carry away is that the discrepancy grows with the width of the band.
What Q Actually Measures
Quality factor has three definitions that all give the same number, and each is illuminating.
As selectivity, Q is centre frequency over bandwidth, which is what this page computes. As energy storage, Q is 2π times the energy stored in a resonator divided by the energy lost per cycle, which is why a high-Q system rings for a long time. As damping, Q is the reciprocal of twice the damping ratio, which is how the same behaviour appears in a mechanical or control system.
The ringing view is the most useful intuition. A resonator's amplitude decays to about 4 per cent of its initial value after Q cycles, so a Q of 10 rings for about ten cycles and a Q of 1,000 rings for a thousand. That is why a struck wine glass sustains and a struck cushion does not, and why a high-Q filter has a long impulse response and therefore a slow settling time.
That last point is the engineering trade-off. Selectivity and speed pull against each other: a filter narrow enough to reject an adjacent channel is also slow enough to smear a fast transient. There is no way around it, because both are the same property viewed in different domains. The damping ratio calculator covers the time-domain side, and the LC resonant frequency calculator covers where a resonance sits given the components.
Where This Sits Next to the Filter Tools
Three tools on this site touch the same subject from different directions, and knowing which one you want saves time.
The filter cutoff calculator works from components: give it a resistance and a capacitance and it tells you where the corner frequency lands. The LC resonant frequency calculator does the same for an inductor and a capacitor at resonance. Both start from hardware and produce a frequency.
This page starts from frequencies and produces a description of the band. It does not know or care what produced the band, which means it works equally well for an electronic filter, an acoustic resonance, a mechanical mode, an optical cavity or a mode in a control loop. If you have measured two half-power points on anything at all, this is the page that turns them into Q.
For related frequency arithmetic, the frequency period calculator converts between frequency and period, and the frequency converter handles unit rescaling.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Averaging the cutoffs to get the centre — the centre of a resonant band is the geometric mean. The arithmetic midpoint is close for a narrow band and badly wrong for a wide one.
- Using cutoffs measured at the wrong level — bandwidth and Q are defined at the half-power points, 3 dB down. Passband edges quoted at 1 dB or 0.5 dB give a narrower band and inflate Q.
- Confusing signal bandwidth with data rate — both are called bandwidth. This page deals in hertz. Bits per second is a different quantity with a different tool.
- Quoting Q for a very wide band — below about one, Q stops being informative. Octave span or fractional bandwidth describes a wide band far better.
- Assuming high Q is always better — a high-Q filter rings, settles slowly and is sensitive to component tolerance. Selectivity is bought with time-domain performance.
Related Free Tools From Arb Digital
To get a cutoff from components, use the filter cutoff calculator or the LC resonant frequency calculator. For the time-domain view of the same behaviour, use the damping ratio calculator. Convert between frequency and period with the frequency period calculator and between units with the frequency converter. If you meant data throughput rather than a span of frequencies, the bandwidth converter, the bandwidth requirement calculator and the bandwidth delay product calculator are the tools you want. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
Subtract the lower cutoff from the upper one. Both should be half-power points, where the response has fallen 3 decibels from its peak. A band from 950 to 1,050 hertz has a bandwidth of 100 hertz.
No, it is the geometric mean, the square root of their product. For a narrow band the two are almost identical, but for a wide band they differ substantially. A band from 100 to 10,000 hertz has a geometric centre of 1,000 hertz and an arithmetic midpoint of 5,050.
How selective a resonance is, measured as centre frequency divided by bandwidth. A high Q means a narrow band relative to where it sits and a sharp peak. It also equals two pi times the stored energy divided by the energy lost per cycle, so a high Q system rings for many cycles.
The bandwidth divided by the centre frequency, usually written as a percentage. It is exactly the reciprocal of Q, so a Q of 10 is a fractional bandwidth of 10 per cent. It is the more natural description when comparing bands at very different centre frequencies.
Use the exact relations rather than adding and subtracting half the bandwidth. Multiply the centre frequency by the square root of one plus one over four Q squared, then subtract or add one over two Q times the centre frequency. The approximation is only close at high Q.
The damping ratio is one divided by twice Q. A Q of 10 corresponds to a damping ratio of 0.05, which is lightly damped and rings. A Q of 0.5 corresponds to a damping ratio of 1, which is critical damping and the fastest response without overshoot.
No. This page deals with a span of frequencies measured in hertz. Network bandwidth is a data rate in bits per second. The two are related through information theory, but they are different quantities and are calculated in completely different ways.
This tool is provided for educational and study use. It applies the standard half-power definitions of bandwidth, centre frequency and quality factor to a single resonant band, and does not model filter order, ripple, group delay or component tolerance, so treat its output as a design starting point rather than a specification.
