Every digital communication system has to answer two questions: how much spectrum do I need? (occupied bandwidth) and how fast can I theoretically transmit through it? (Shannon capacity). This signal bandwidth calculator resolves the first question for ASK, FSK, PSK, and QAM modulation; the answer depends on the modulation scheme and roll-off factor. Nyquist gave the lower bound on bandwidth; Shannon gave the upper bound on capacity, and the two bounds together define the design space for every radio link.
Nyquist’s bandwidth theorem
For a noiseless channel, the minimum bandwidth required to transmit at symbol rate Rs without inter-symbol interference is Rs/2 Hz (baseband) or Rs Hz (passband). This is the "Nyquist bandwidth." Real systems can't use ideal brick-wall filters, so they add excess bandwidth via a roll-off factor α (typically 0.20–0.35) using a raised cosine pulse-shaping filter, giving occupied bandwidth B = Rs · (1 + α).
Shannon’s channel capacity
Shannon's 1948 formula: C = B · log2(1 + SNR), where SNR is in linear scale (not dB). This is the upper bound on error-free bit rate for any modulation, any coding. At SNR = 20 dB (100×), spectral efficiency η = log2(101) ≈ 6.66 bits/s/Hz — meaning a 1 MHz channel could theoretically carry up to 6.66 Mbps. Real systems get close (LTE, WiFi 6 reach ~80% of Shannon η in good conditions) but never exceed it. Eb/N0 (energy per bit over noise density) is the normalized counterpart: it stays fixed regardless of bandwidth, making it the standard figure of merit for comparing PSK bandwidth, QAM bandwidth, and FSK bandwidth on equal footing. OFDM-based systems like LTE and WiFi subdivide the channel into many narrow sub-carriers and apply the Shannon limit per sub-carrier independently, allowing the water-filling power allocation to approach the theoretical bound more closely in frequency-selective fading channels.
Bits per symbol — the modulation tradeoff
Higher-order modulations pack more bits per symbol: BPSK = 1, QPSK = 2, 16-QAM = 4, 256-QAM = 8, 4096-QAM = 12. Bandwidth stays the same per symbol — you get more bits without more spectrum. But the constellation points get closer together, so you need higher SNR to distinguish them. Rule of thumb: each extra bit/symbol needs ~3 dB more SNR. 1024-QAM (WiFi 6) needs ~30 dB SNR; 4096-QAM (WiFi 7) needs ~36 dB. Gray coding of the constellation maps adjacent points to bit patterns differing by only one bit, so the most common error (slipping one point) costs only a single bit error rather than log₂(M) bits.
ASK, FSK, PSK, QAM
- ASK (Amplitude Shift Keying): encodes data in signal amplitude. OOK (on-off keying) is the simplest. Vulnerable to amplitude noise. Used in optical links and simple RFID systems.
- FSK (Frequency Shift Keying): encodes data in carrier frequency. Constant-envelope (immune to amplitude noise), used in Bluetooth Classic, garage door openers, low-rate IoT. Wider spectrum than PSK/QAM. FSK bandwidth depends on the modulation index
h = 2Δf / Rs; this tool's 2-FSK/MSK/4-FSK values assume h = 0.5 (the most spectrally efficient case — MSK, Minimum Shift Keying). For larger h, the actual bandwidth is wider (Carson's rule: B ≈ 2Δf + Rs). Gaussian FSK (GFSK), used in Bluetooth, pre-filters the frequency deviation with a Gaussian pulse to reduce out-of-band emissions.
- PSK (Phase Shift Keying): encodes data in carrier phase. BPSK (2 phases), QPSK (4 phases), 8-PSK (8 phases). Constant envelope — excellent for power-amplifier linearity requirements. Limited to k ≤ ~3 before SNR demand becomes prohibitive. You can convert RF carrier frequency to wavelength to size your antenna for any PSK link.
- QAM (Quadrature Amplitude Modulation): encodes data in BOTH phase and amplitude. The constellation is a 2D grid of points. Far more bits per symbol than PSK at the same SNR. Used in LTE, WiFi 4+, DOCSIS, DVB, and cable broadband systems.
What's the difference between Nyquist bandwidth and occupied bandwidth?
Nyquist bandwidth is the THEORETICAL minimum needed to transmit at symbol rate Rs with no inter-symbol interference — equal to Rs at passband (or Rs/2 baseband). Occupied bandwidth is what your signal ACTUALLY uses after pulse shaping. Real systems use raised-cosine (or root-raised-cosine) filters with roll-off α > 0, giving occupied B = Rs · (1 + α). For α = 0.25, the occupied bandwidth is 25% wider than Nyquist. The smaller α, the more efficient — but the filter gets harder to build and more sensitive to timing errors.
Why can't real systems reach the Shannon limit?
Shannon proved capacity exists with arbitrarily small error probability, but achieving it requires (a) infinitely long codewords (infinite latency), (b) infinite-complexity decoders, and (c) Gaussian-distributed signal (continuous, not discrete constellations). Real systems use finite-length error-correction codes (LDPC, polar, turbo), discrete QAM constellations, and have practical hardware impairments. Modern systems get 70–85% of Shannon η; the last few % requires exotic coding and is often not worth the complexity. 5G NR LDPC + 256-QAM in good conditions reaches ~80%.
What is Eb/N0 and why does it matter?
Eb/N0 (read "E-b over N-naught") is the energy per bit divided by noise power spectral density — a normalized SNR that lets you compare systems with different data rates and bandwidths fairly. Relationship: Eb/N0 = SNR × B / Rb = SNR / η. Lower Eb/N0 means more efficient use of energy per bit. Each modulation has a minimum Eb/N0 to achieve a target bit-error rate (BER). BPSK needs ~9.6 dB for 10⁻⁵ BER (uncoded); 256-QAM needs ~24 dB. Error-correction codes can reduce required Eb/N0 by 3–10 dB ("coding gain").
Why does QAM beat PSK for high bit rates?
At high orders, QAM's 2D constellation packs points more efficiently than PSK's 1D ring. Consider 16-QAM (4×4 grid) vs 16-PSK (16 points on a circle). The 16 points on a circle are very close together — small noise pushes you to a neighbor. The 16-point grid has more breathing room because the constellation expands in both dimensions. For a given average power, QAM's minimum point spacing is larger than PSK's for M ≥ 16. Below M = 8, the difference is minor; above it, QAM dominates. That's why no one uses 16-PSK or 32-PSK in practice.
What's a typical real-world example of these numbers?
5G NR with 100 MHz channel: typical operating point is 256-QAM with rate-3/4 LDPC at ~25 dB SNR. Symbol rate ~100 Mbaud × spectral-efficiency overhead ≈ 90 Mbaud net. Bit rate = 90 Mbaud × 8 bits/symbol × 0.75 (code rate) ≈ 540 Mbps. Shannon at 25 dB, 100 MHz = ~830 Mbps. So 5G achieves ~65% of Shannon η — a typical real-world figure. The remaining 35% is overhead for control channels, pilot signals, guard bands, and coding inefficiency.
Should I use a smaller roll-off α to save bandwidth?
Tradeoffs. Smaller α (closer to 0) means closer to Nyquist minimum — saves spectrum. But: (a) the filter needs more taps to achieve sharper roll-off, increasing DSP complexity and latency; (b) the time-domain pulse has bigger sidelobes, more sensitive to timing jitter; (c) the peak-to-average power ratio (PAPR) grows, requiring more linear amplifiers. Modern systems pick α based on the regulatory mask and hardware budget. LTE uses α ≈ 0.22, DVB-S2X allows 0.05–0.35. Don't go below 0.10 unless you have a very stable clock and high-quality DSP.
What is the difference between a raised-cosine filter and a root-raised-cosine (RRC) filter?
A raised-cosine (RC) filter achieves zero inter-symbol interference (ISI) at the receiver. In practice the total RC response is split equally between transmitter and receiver as two matched root-raised-cosine (RRC) filters — each with a transfer function that is the square root of the RC. The transmitter RRC shapes the pulse to limit bandwidth; the receiver RRC matches the transmitter to maximize SNR and, together, the pair reproduces the RC response that eliminates ISI. This calculator plots the full raised-cosine spectrum, which represents what you would observe on a spectrum analyser for the combined transmit filter.
How does OFDM bandwidth compare to the single-carrier formulas here?
OFDM (Orthogonal Frequency-Division Multiplexing) divides the channel into N sub-carriers, each a narrow-band single-carrier QAM or PSK signal. The total occupied bandwidth is approximately N × sub-carrier spacing × (1 + guard-band overhead). Each sub-carrier individually follows the Nyquist/Shannon maths this tool computes; the aggregate behaves as if the entire channel bandwidth is filled. LTE 20 MHz, for example, uses 1200 sub-carriers at 15 kHz spacing for 18 MHz of occupied signal, with the remainder as guard band. This calculator's single-carrier result gives the sub-carrier bandwidth, not the full OFDM channel width.
How do I calculate FSK bandwidth using Carson's rule when the modulation index is not 0.5?
Carson's rule for FSK occupied bandwidth is B ≈ 2Δf + Rs, where Δf is the peak frequency deviation and Rs is the symbol rate. The modulation index h = 2Δf / Rs, so Δf = h × Rs / 2, giving B ≈ h × Rs + Rs = Rs × (1 + h). This tool assumes h = 0.5 (MSK) for maximum spectral efficiency. If your system uses h = 1 (standard 2-FSK), the bandwidth is twice the symbol rate — double the MSK figure. For GFSK (Bluetooth BLE, h ≈ 0.5, BT = 0.5), the pre-Gaussian filtering slightly narrows the spectral main lobe but the Carson's-rule estimate is still a useful starting point. For precise out-of-band emission calculations, use the actual power spectral density formula or a link with your
WiFi channel bandwidth reference to cross-check regulatory limits.