Ferkans — Interactive Telecom Tutor

How Much Noise Does a Filter Let Through?

We know that a filter shapes the noise PSD. But for engineering calculations — link budgets, receiver sensitivity specifications, noise figure cascading — we need a single number that captures "how much noise this filter admits." The noise equivalent bandwidth provides exactly this: it is the bandwidth of an ideal rectangular filter that would pass the same total noise power. This seemingly simple definition turns out to be remarkably useful: it lets us replace messy integrals over $|\check{h}(f)|^2$ with a single multiplication by $B_N$ .

Definition:
Noise Equivalent Bandwidth

The noise equivalent bandwidth of an LTI filter with frequency response $\check{h}(f)$ is $B_N = \frac{\int_{-\infty}^{\infty} |\check{h}(f)|^2\, df}{2\, |\check{h}(f_{\max})|^2} = \frac{1}{2} \cdot \frac{\int_{-\infty}^{\infty} |\check{h}(f)|^2\, df}{|\check{h}(f_{\max})|^2},$ where $f_{\max}$ is the frequency of maximum gain (typically $f_{\max} = 0$ for a lowpass filter, giving $|\check{h}(0)|^2$ in the denominator). The factor of 2 makes $B_N$ a one-sided bandwidth.

Equivalently, $B_N$ is the bandwidth of an ideal rectangular filter with the same peak gain that passes the same total white-noise power: $\int_{-\infty}^{\infty} |\check{h}(f)|^2\, df = 2\, |\check{h}(f_{\max})|^2 \cdot B_N.$

For a lowpass filter with $\check{h}(0) = 1$ , the output noise power due to white noise of PSD $N_0/2$ is simply $\mathcal{P}_N = N_0 B_N$ .

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Definition:
Noise Figure

The noise figure $F$ of a two-port device at temperature $T_0 = 290$ K is the ratio of the actual output noise power to the output noise power that would be present if the device were noiseless: $F = \frac{P_{N,\text{out}}}{G \cdot k_B T_0 B_N},$ where $G$ is the power gain, $k_B$ is Boltzmann's constant, and $B_N$ is the noise equivalent bandwidth. Equivalently, $F = \text{SNR}_{\text{in}} / \text{SNR}_{\text{out}}$ : the noise figure measures the SNR degradation caused by the device.

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Theorem: Output Noise Power via Noise Bandwidth

If white noise with two-sided PSD $N_0/2$ passes through a filter with peak gain $|\check{h}(f_{\max})|^2 = G_0$ and noise equivalent bandwidth $B_N$ , the output noise power is $\mathcal{P}_N = \frac{N_0}{2} \int |\check{h}(f)|^2\, df = N_0\, G_0\, B_N.$ For a unity-gain lowpass filter ( $G_0 = 1$ ), this simplifies to $\mathcal{P}_N = N_0 B_N$ .

The noise bandwidth converts the problem of integrating $|\check{h}(f)|^2$ into a single multiplication. We pretend the filter is an ideal brick-wall of bandwidth $B_N$ and peak gain $G_0$ — the noise power is the same as the actual filter's.

Proof

Direct from definition

By definition of $B_N$ : $\int |\check{h}(f)|^2\, df = 2 G_0 B_N$ . The output noise power is $(N_0/2) \cdot 2 G_0 B_N = N_0 G_0 B_N$ .

Example: Noise Bandwidth of an RC Lowpass Filter

Compute the noise equivalent bandwidth of the RC lowpass filter with frequency response $\check{h}(f) = \frac{1}{1 + j 2\pi f RC}$ , and compare it to the 3-dB bandwidth.

Solution

Squared magnitude

$|\check{h}(f)|^2 = \frac{1}{1 + (2\pi f RC)^2}$ , with peak $|\check{h}(0)|^2 = 1$ .

Integral

$\int_{-\infty}^{\infty} \frac{df}{1 + (2\pi f RC)^2} = \frac{1}{2\pi RC} \cdot \pi = \frac{1}{2RC}.$ $

Noise bandwidth

$B_N = \frac{1}{2} \cdot \frac{1/(2RC)}{1} = \frac{1}{4RC}.$ $

Compare to 3-dB bandwidth

The 3-dB bandwidth is $f_{3\text{dB}} = \frac{1}{2\pi RC}$ . So $B_N = \frac{\pi}{2} f_{3\text{dB}} \approx 1.57\, f_{3\text{dB}}$ . The noise bandwidth is about 57% wider than the 3-dB bandwidth — the slow rolloff of the single-pole filter lets through extra noise at high frequencies.

Example: Noise Bandwidth of an Ideal Lowpass Filter

Show that the noise equivalent bandwidth of an ideal lowpass filter with cutoff $W$ Hz equals $W$ .

Solution

Frequency response

$\check{h}(f) = \text{rect}(f/(2W))$ , so $|\check{h}(f)|^2 = 1$ for $|f| \leq W$ and $0$ otherwise. Peak gain: $|\check{h}(0)|^2 = 1$ .

Compute $B_N$

$\int |\check{h}(f)|^2\, df = 2W$ . So $B_N = \frac{1}{2} \cdot \frac{2W}{1} = W$ . The ideal filter is the reference: its noise bandwidth equals its actual bandwidth.

Example: Noise Bandwidth of a Butterworth Filter

The $n$ -th order Butterworth lowpass filter has $|\check{h}(f)|^2 = \frac{1}{1 + (f/f_c)^{2n}}$ . Find $B_N$ as a function of $n$ and $f_c$ , and show it approaches $f_c$ as $n \to \infty$ .

Solution

Integral

$\int_{-\infty}^{\infty} \frac{df}{1 + (f/f_c)^{2n}} = 2 f_c \int_0^{\infty} \frac{du}{1 + u^{2n}} = 2 f_c \cdot \frac{\pi}{2n \sin(\pi/(2n))}.$ $ The last step uses the beta function identity.

Noise bandwidth

$B_N = \frac{1}{2} \cdot 2 f_c \cdot \frac{\pi}{2n \sin(\pi/(2n))} = \frac{\pi f_c}{2n \sin(\pi/(2n))}.$ $

Special cases and limit

$n = 1$ : $B_N = \frac{\pi f_c}{2} \approx 1.571 f_c$ (the RC filter).
$n = 2$ : $B_N = \frac{\pi f_c}{2\sqrt{2}} \approx 1.111 f_c$ .
$n \to \infty$ : $\sin(\pi/(2n)) \approx \pi/(2n)$ , so $B_N \to f_c$ . Higher-order Butterworth filters approach the ideal filter's noise bandwidth.

Noise Bandwidth for Different Filter Shapes

Compare the noise equivalent bandwidth $B_N$ across filter types and orders. The plot shows $|\check{h}(f)|^2$ overlaid with the equivalent ideal rectangular filter of width $B_N$ and the same peak gain.

Parameters

Filter type

Filter order

n

2

f_c

(Hz)3

Noise Bandwidth Ratio $B_N / f_{3\text{dB}}$ for Common Filters

Filter	$B_N / f_c$	$B_N / f_{3\text{dB}}$	Notes
Ideal lowpass	1.000	1.000	Reference — no excess noise
RC (1st order Butterworth)	$\pi/2 \approx 1.571$	$\pi/2 \approx 1.571$	57% excess noise bandwidth
2nd order Butterworth	1.111	1.222	Better rolloff reduces excess
4th order Butterworth	1.026	1.084	Approaching ideal
Gaussian	$\sqrt{\pi/2} \approx 1.253$	$\sqrt{\pi \ln 2} \approx 1.476$	No ringing, moderate excess

Quick Check

As the order $n$ of a Butterworth lowpass filter increases, what happens to the ratio $B_N / f_c$ ?

It approaches 1 (ideal rectangular filter)

It approaches $\pi/2$

It increases without bound

Correction:

It approaches 1 (ideal rectangular filter)

Higher-order Butterworth filters have sharper rolloff, approaching the ideal brick-wall response. Their noise bandwidth approaches the actual cutoff frequency.

⚠️Engineering Note

Friis Formula for Cascaded Noise Figures

For $k$ stages in cascade with noise figures $F_1, F_2, \ldots, F_k$ and power gains $G_1, G_2, \ldots, G_k$ , the overall noise figure is $F_{\text{total}} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots + \frac{F_k - 1}{\prod_{i=1}^{k-1} G_i}.$ The first stage dominates: a low-noise amplifier (LNA) at the front end can make the contributions of subsequent stages negligible. This is why every wireless receiver places the LNA immediately after the antenna.

Practical Constraints

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Each stage must be impedance-matched for the formula to apply
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The formula assumes each stage adds noise independently

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🔧Engineering Note

Equivalent Noise Temperature

An alternative to noise figure is the equivalent noise temperature $T_e$ , related by $F = 1 + T_e / T_0$ where $T_0 = 290$ K. Noise temperature is preferred for low-noise systems (radio astronomy, deep-space communications) where the noise figure is very close to 1 and the dB value loses resolution. A cryogenically cooled LNA might have $T_e = 15$ K ( $F = 1.05$ , or $0.22$ dB), which is more informatively expressed as a temperature than as a noise figure.

Common Mistake: Using 3-dB Bandwidth Instead of Noise Bandwidth

Mistake:

Computing noise power as $N_0 \times f_{3\text{dB}}$ for a non-ideal filter.

Correction:

The 3-dB bandwidth and the noise bandwidth are different for all real filters. The noise power is $N_0 B_N$ , not $N_0 f_{3\text{dB}}$ . For a single-pole RC filter, using $f_{3\text{dB}}$ instead of $B_N$ underestimates the noise power by a factor of $\pi/2 \approx 1.57$ , which is a 2 dB error.

Why This Matters: Receiver Sensitivity and the Noise Floor

The noise floor of a receiver is $P_N = k_B T_0 B_N F$ , where $k_B T_0 = -174$ dBm/Hz at room temperature, $B_N$ is the noise equivalent bandwidth, and $F$ is the noise figure. The receiver sensitivity — the minimum detectable signal power — is $P_{\min} = P_N \cdot \text{SNR}_{\min}$ . Every dB of noise figure improvement or bandwidth reduction directly improves sensitivity. This is why 5G NR receivers specify noise figures below 5 dB and why narrowband IoT protocols use bandwidths as small as 180 kHz.

Historical Note: Harald Friis and the Noise Figure

1940s

The concept of noise figure was formalized by Harald T. Friis of Bell Labs in 1944. His cascade formula showed that the first amplifier stage dominates the overall noise performance — a result that shaped the entire architecture of radio receivers. Before Friis, engineers used ad hoc measures of receiver quality. The noise figure provided a universal, manufacturer-independent specification that enabled modular receiver design.

Noise Equivalent Bandwidth

The bandwidth $B_N$ of an ideal rectangular filter with the same peak gain that produces the same output noise power when driven by white noise. For a lowpass filter with unity DC gain: $B_N = \frac{1}{2} \int |\check{h}(f)|^2\, df$ .

Related: Noise Figure

Noise Figure

The ratio $F = \text{SNR}_{\text{in}} / \text{SNR}_{\text{out}}$ measuring the SNR degradation caused by a device. Equivalently, $F = 1 + T_e/T_0$ where $T_e$ is the equivalent noise temperature.

Related: Noise Equivalent Bandwidth

Key Takeaway

The noise equivalent bandwidth $B_N$ reduces the output noise power calculation to a single number: $\mathcal{P}_N = N_0 B_N$ for a unity-gain filter in white noise. For real filters, $B_N > f_{3\text{dB}}$ because the non-ideal rolloff lets through extra noise. Higher-order filters have $B_N$ closer to their cutoff frequency. The noise figure $F$ extends this to characterize noisy devices, and Friis's cascade formula shows that the first stage dominates overall noise performance.

Noise Bandwidth and Equivalent Noise

How Much Noise Does a Filter Let Through?

Definition: Noise Equivalent Bandwidth

Definition: Noise Figure

Theorem: Output Noise Power via Noise Bandwidth

Direct from definition

Example: Noise Bandwidth of an RC Lowpass Filter

Squared magnitude

Integral

Noise bandwidth

Compare to 3-dB bandwidth

Example: Noise Bandwidth of an Ideal Lowpass Filter

Frequency response

Compute $B_N$

Example: Noise Bandwidth of a Butterworth Filter

Integral

Noise bandwidth

Special cases and limit

Noise Bandwidth for Different Filter Shapes

Parameters

Noise Bandwidth Ratio BN/f3dBB_N / f_{3\text{dB}}BN​/f3dB​ for Common Filters

Quick Check

Friis Formula for Cascaded Noise Figures

Equivalent Noise Temperature

Common Mistake: Using 3-dB Bandwidth Instead of Noise Bandwidth

Why This Matters: Receiver Sensitivity and the Noise Floor

Historical Note: Harald Friis and the Noise Figure

Noise Equivalent Bandwidth

Noise Figure

Key Takeaway

Definition:
Noise Equivalent Bandwidth

Definition:
Noise Figure

Noise Bandwidth Ratio $B_N / f_{3\text{dB}}$ for Common Filters