Why is noise bandwidth always wider than 3 dB bandwidth?

The 3 dB bandwidth measures the frequency span between the points where the filter gain drops to half its peak power (minus 3 dB). But the filter still passes noise beyond the 3 dB points: the skirts of the filter response contribute additional noise power that a rectangular filter of the same width would not. The noise bandwidth accounts for this extra noise by defining a wider rectangular equivalent. For a single-pole RC filter, the noise bandwidth is pi/2 = 1.57 times the 3 dB bandwidth. For a 4th-order Butterworth, the ratio drops to 1.026 because the steeper skirts contribute less out-of-band noise. As the filter order increases toward infinity (approaching an ideal brick-wall response), the noise bandwidth approaches the 3 dB bandwidth. Using 3 dB bandwidth instead of noise bandwidth in the kTB formula underestimates the noise power, which causes the sensitivity calculation to be optimistic by the ratio factor in dB.

How is noise bandwidth measured or calculated?

Analytically, the noise bandwidth is B_n = (1/|H_max|^2) times the integral from 0 to infinity of |H(f)|^2 df, where H(f) is the filter's voltage transfer function. For simple filter types, closed-form solutions exist. For complex or measured filters, numerical integration of the measured frequency response is used: sweep the filter on a VNA, export the S21 magnitude data, square each point, sum them all up weighted by the frequency step, and divide by the peak squared magnitude. Spectrum analyzers specify their noise bandwidth directly in their specifications because it is needed to convert displayed noise levels to noise density. A typical spectrum analyzer with a specified 10 kHz resolution bandwidth (3 dB) might have a noise bandwidth of 11.5 kHz, and the instrument applies this correction factor internally when measuring noise.

When does the noise bandwidth vs 3 dB bandwidth difference actually matter?

For high-order filters (5th order and above), the noise bandwidth is within 5 percent of the 3 dB bandwidth, and the error from using B_3dB in the kTB formula is less than 0.2 dB. This is negligible for most engineering calculations. The difference matters significantly in two scenarios. First, for single-pole or low-order filters: a single-pole RC filter has a noise bandwidth 1.57 times its 3 dB bandwidth, a 1.96 dB error. An IF amplifier with a single-tuned circuit as its selectivity element would have its sensitivity overestimated by nearly 2 dB if the 3 dB bandwidth were used. Second, for precision noise measurements: when measuring noise figure or noise temperature with a noise source and a spectrum analyzer, the noise bandwidth correction must be applied accurately (to 0.1 dB) to get meaningful results.

Fundamentals

Noise Bandwidth

A receiver's IF filter has a 3 dB bandwidth of 200 kHz. The designer calculates the noise floor using kTB: −174 + 10·log(200,000) = −174 + 53 = −121 dBm. But the actual noise power is higher because the filter skirts pass noise beyond the 3 dB points. The noise bandwidth, the equivalent rectangular width that captures the same total noise power, is 230 kHz (1.15× wider). The correct noise floor is −174 + 10·log(230,000) = −120.4 dBm: 0.6 dB worse than the optimistic estimate. For a single-pole filter, the error is 2 dB. For precise sensitivity calculations, particularly with low-order filters, noise bandwidth must replace 3 dB bandwidth in every kTB computation.

Category: Fundamentals

Symbol: B_n

Rule: B_n ≥ B_3dB (always)

Noise Bandwidth Ratio by Filter Type

Filter Type	B_n/B_3dB	dB Error	Notes
Single-pole (RC)	1.571 (π/2)	1.96 dB	Worst case for common filters
2nd-order Butterworth	1.111	0.46 dB	Two cascaded single-tuned circuits
4th-order Butterworth	1.026	0.11 dB	Negligible for most applications
5th-order Chebyshev 0.5 dB	1.038	0.16 dB	Slight undershoot from ripple
Gaussian (matched filter)	1.065	0.27 dB	Spectrum analyzer default
Ideal rectangular	1.000	0 dB	Not physically realizable

Noise bandwidth definition:
B_n = (1/|H_max|²) × ∫₀^∞ |H(f)|² df

Correct noise power:
N = kT·B_n (not kT·B_3dB)

Sensitivity error from using B_3dB:
Δ = 10·log(B_n/B_3dB) dB
Single-pole: Δ = 10·log(1.571) = 1.96 dB optimistic

Common Questions

Frequently Asked Questions

Why always wider than B_3dB?

Filter skirts pass noise beyond the −3 dB points. A rectangular filter of B_3dB width would not. B_n accounts for this extra noise. Single-pole: 1.57×. 4th-order Butterworth: 1.026×. Approaches 1.0 as order → ∞.

How measured?

VNA sweep: export |S₂₁|, square each point, integrate (sum × Δf), divide by peak². Spectrum analyzers specify B_n directly (e.g., 10 kHz RBW has 11.5 kHz noise BW) and apply correction internally.

When does it matter?

High-order filters (≥5th): <0.2 dB error, negligible. Single-pole: 2 dB error, significant. Precision noise measurements (NF, Te): must correct to 0.1 dB accuracy. Calibrated noise sources specify their ENR assuming exact noise bandwidth application, so measurement errors propagate directly into reported NF values. When budgeting system sensitivity, always use B_n from the filter datasheet rather than the 3 dB specification.

Related Terms

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