Figure 1: Bandwidth is the frequency range between the −3 dB cutoff points. It determines data capacity (Shannon theorem), rise time, and filter selectivity (Q factor).
Table of Contents
What Is Bandwidth?
Bandwidth is the range of frequencies a system can handle, measured between the −3 dB points where the signal drops to 70.7% of peak voltage (half power). A 100 MHz bandwidth system passes signals from fL to fH where fH − fL = 100 MHz. Signals outside this range are attenuated. The Decibel Calculator explains the −3 dB point in detail — it is the universal definition of cutoff in electronics.
The calculator above has three modes. Mode 1 computes bandwidth from cutoff frequencies. Mode 2 derives bandwidth from rise time. Mode 3 finds bandwidth from centre frequency and Q factor. Every mode computes all nine derived quantities, so you always get the complete picture regardless of your starting point.
Nine Results from Any Input
fc = √(fL × fH) — centre frequency (geometric mean)
Q = fc / BW — quality factor (selectivity)
tr = 0.35 / BW — rise time (10–90%)
Fractional BW = (BW / fc) × 100%
Octaves = log₂(fH / fL)
Shannon = BW × log₂(1 + SNR) — max data rate at 30 dB SNR
Nyquist = 2 × BW — max symbol rate for binary signalling
fL / fH — the actual cutoff frequencies
Mode 1 — From Frequencies
Enter lower cutoff (fL) and upper cutoff (fH) in Hz, kHz, MHz, or GHz. These are the −3 dB points of your system’s frequency response.
Example: Audio Band (20 Hz – 20 kHz)
BW = 20,000 − 20 = 19,980 Hz ≈ 20 kHz
fc = √(20 × 20,000) = 632.5 Hz
Q = 632.5 / 19,980 = 0.032 — extremely wideband
Octaves = log₂(1000) = 9.97 ≈ 10 octaves
Shannon (30 dB) ≈ 199 kbps
The audio band spans almost exactly 10 octaves. Q of 0.032 confirms it is an extremely wideband system, not a resonant filter.
Example: Wi-Fi 2.4 GHz (2.4 – 2.5 GHz)
BW = 100 MHz
fc = 2.449 GHz
Fractional BW = 100/2449 × 100 = 4.08% — narrowband
tr = 0.35 / 100 MHz = 3.5 ns
Shannon (30 dB) ≈ 997 Mbps
Shannon’s limit of ~1 Gbps at 30 dB SNR explains why 802.11n/ac use wider channels (40/80/160 MHz) and MIMO to push throughput higher — they are approaching the theoretical ceiling of the available bandwidth.
Mode 2 — From Rise Time
Enter the 10–90% rise time (tr) in ns, µs, or ms. The calculator derives bandwidth using BW ≈ 0.35 / tr, which is valid for Gaussian response systems like oscilloscopes and most amplifiers. This is the essential formula for oscilloscope selection and digital signal integrity.
Example: 1 ns Rise Time (Fast Logic)
BW = 0.35 / 1 ns = 350 MHz
Shannon (30 dB) ≈ 3.49 Gbps
Choose a 500 MHz oscilloscope for 3× margin. The scope’s own rise time (0.7 ns at 500 MHz) combines with the signal as tmeasured = √(tsignal² + tscope²), so a scope with similar bandwidth underreports the true rise time. The Signal Attenuation Calculator shows how cable loss at these high frequencies further degrades edge speed.
Example: 10 ns Rise Time
BW = 0.35 / 10 ns = 35 MHz
A standard 100 MHz oscilloscope handles this with plenty of margin.
Mode 3 — From Q Factor
Enter centre frequency (fc) and quality factor (Q). The calculator derives bandwidth from BW = fc / Q. Higher Q means narrower bandwidth relative to centre frequency — a more selective filter.
Example: 1 kHz, Q = 10
BW = 1,000 / 10 = 100 Hz
fL = 950 Hz | fH = 1,050 Hz
tr = 0.35 / 100 = 3.5 ms
Fractional BW = 10%
Moderately selective. Suitable for audio tone detection, band-pass filtering of a specific frequency, or a tuned amplifier stage.
Example: 455 kHz IF Filter, Q = 100
BW = 455,000 / 100 = 4,550 Hz ≈ 4.55 kHz
fL = 452.7 kHz | fH = 457.3 kHz
Fractional BW = 1%
The classic AM radio IF filter. Q of 100 gives a 4.55 kHz passband — just wide enough for voice audio but narrow enough to reject the adjacent channel 10 kHz away. Crystal filters achieve Q values above 10,000 for even narrower passbands (SSB, CW). The Gain Calculator can then size the IF amplifier stage that follows the filter, accounting for the gain-bandwidth product limitation.
Rise Time and Bandwidth
BW ≈ 0.35 / tr links the frequency domain (bandwidth) to the time domain (rise time). A system with 100 MHz bandwidth produces edges no faster than 3.5 ns. A signal with 1 ns edges contains frequency content up to 350 MHz. This relationship drives oscilloscope selection: the scope must have enough bandwidth to faithfully reproduce the signal’s rise time.
The 0.35 constant applies to Gaussian response systems (most amplifiers and oscilloscopes). For Bessel or Butterworth filters the constant differs slightly (0.34 and 0.38 respectively), but 0.35 is a safe general approximation.
Q Factor Explained
Q = fc / BW measures selectivity. High Q (narrow bandwidth) means the filter passes a tight range around the centre frequency and rejects everything else. Low Q (wide bandwidth) means the response is broad. Q also relates to energy storage: a high-Q resonator stores energy for many cycles before dissipating it, which is why crystal oscillators (Q > 10,000) maintain frequency so precisely.
| Q Range | Description | Typical Use |
|---|---|---|
| < 1 | Ultra-wideband | Full audio range, broadband amplifiers |
| 1 – 10 | Wideband | Audio equalisers, wideband filters |
| 10 – 100 | Moderate selectivity | IF filters, tuned circuits, tone detection |
| 100 – 1,000 | Narrow band | Crystal filters, narrow IF stages |
| > 10,000 | Extremely narrow | Crystal oscillators, atomic clocks |
Shannon Capacity
C = BW × log₂(1 + SNR) is the theoretical maximum data rate for a channel with a given bandwidth and signal-to-noise ratio. At 30 dB SNR (1000:1), each hertz of bandwidth supports approximately 10 bits per second. This is a hard ceiling — no modulation scheme can exceed it. Real systems achieve 50–90% of Shannon capacity with modern error-correction coding (LDPC, turbo codes, polar codes).
This is why 5G uses wider frequency bands (higher bandwidth) and MIMO (effectively higher SNR per stream) — both directly increase the Shannon limit. The Duty Cycle Calculator handles a related concept for pulsed signals, where the effective data rate depends on the fraction of time the channel is active.
Fractional Bandwidth
Fractional bandwidth = BW / fc × 100%. It normalises bandwidth to the operating frequency, making it easy to compare systems at different frequencies. A 100 MHz bandwidth at 2.4 GHz is 4.08% (narrowband). The same 100 MHz at 200 MHz is 50% (ultra-wideband). Standard narrowband filter and amplifier designs work well below about 10% fractional bandwidth. Above 20%, wideband design techniques are required.
Frequently Asked Questions
What is bandwidth?
How does rise time relate to bandwidth?
What is Shannon capacity?
What does Q factor mean in practice?
What is fractional bandwidth?
Why does the calculator show Shannon capacity at 30 dB SNR?
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