---
title: "Moments of random variable; NEB of RC-LPF; output noise power"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:moments-neb-rc-lpf"
source: https://www.bctengineer.com/q/semester_6/communication_systems/moments-neb-rc-lpf
site: BCT Engineering
---

# Moments of random variable; NEB of RC-LPF; output noise power

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2071 Chaitra (CS II), 5+1 marks
- Video: [Noise Equivalent Bandwidth](https://www.youtube.com/watch?v=ikooxwhIIr8) — Anoop C V
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/moments-neb-rc-lpf
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

Define moment and central moment of continuous random variable. Show that first central moment is always zero. Determine the noise equivalent bandwidth of RC-LPF and that of ideal LPF of zero frequency response one. Also, find output noise power of this RC-LPF when input is white noise.

## Short answer

The nth moment is E[X^n] = ∫ x^n f_X(x) dx; the nth central moment is E[(X-μ)^n]. First central moment is zero because E[X-μ] = μ - μ = 0. The noise equivalent bandwidth of an RC-LPF is 1/(4RC), for an ideal LPF it equals its bandwidth B. Output noise power for white noise input is N_0/(4RC).

## Answer

#### Moments and central moments

For a continuous random variable $X$ with PDF $f_X(x)$:

- **$n$th moment** (about the origin):
$$
E[X^n] = \int_{-\infty}^{\infty} x^n f_X(x)\,dx
$$
The first moment is the mean $\mu = E[X]$; the second is the mean-square value $E[X^2]$.
- **$n$th central moment** (about the mean):
$$
E[(X-\mu)^n] = \int_{-\infty}^{\infty} (x-\mu)^n f_X(x)\,dx
$$
The second central moment is the variance $\sigma^2 = E[X^2]-\mu^2$.

**First central moment is always zero:**

$$
\begin{aligned}
E[X-\mu] &= \int_{-\infty}^{\infty}(x-\mu)f_X(x)\,dx \\
&= \int_{-\infty}^{\infty}x f_X(x)\,dx - \mu\int_{-\infty}^{\infty}f_X(x)\,dx \\
&= \mu - \mu(1) = 0
\end{aligned}
$$

#### Noise equivalent bandwidth of RC low-pass filter

The NEB $B_N$ is the bandwidth of an ideal LPF, with the same zero-frequency gain $H(0)$, that passes the same white-noise power:

$$
B_N = \frac{1}{|H(0)|^2}\int_0^{\infty}|H(f)|^2\,df
$$

For the RC-LPF, $H(f) = \dfrac{1}{1+j2\pi fRC}$, so $H(0) = 1$ and

$$
\begin{aligned}
B_N &= \int_0^{\infty}\frac{df}{1+(2\pi fRC)^2} \\
&= \frac{1}{2\pi RC}\Big[\tan^{-1}(2\pi fRC)\Big]_0^{\infty} \\
&= \frac{1}{2\pi RC}\cdot\frac{\pi}{2} = \frac{1}{4RC}
\end{aligned}
$$

Since the 3 dB bandwidth is $f_{3dB} = 1/(2\pi RC)$, $B_N = \frac{\pi}{2}f_{3dB} \approx 1.57 f_{3dB}$.

**Ideal LPF** with $|H(f)| = 1$ for $|f| \le B$ and 0 elsewhere:

$$
B_N = \frac{1}{1}\int_0^{B}1\,df = B
$$

So the NEB of an ideal LPF equals its actual bandwidth.

#### Output noise power of the RC-LPF

Input white noise has two-sided PSD $S_X(f) = N_0/2$. Output PSD is $S_Y(f) = |H(f)|^2 N_0/2$, so

$$
\begin{aligned}
P_{out} &= \int_{-\infty}^{\infty}\frac{N_0}{2}\cdot\frac{df}{1+(2\pi fRC)^2} \\
&= \frac{N_0}{2}\cdot 2B_N = \frac{N_0}{2}\cdot\frac{1}{2RC} = \frac{N_0}{4RC}
\end{aligned}
$$

**Answer:** $B_{N,RC} = \dfrac{1}{4RC}$, $B_{N,ideal} = B$, output noise power $= \dfrac{N_0}{4RC}$ (watts, on a 1 $\Omega$ basis).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is E[X - mu] = 0?
   A. Because the PDF is symmetric
   B. Because the integral of x f(x) equals mu and mu times the integral of f(x) is also mu
   C. Because variance is zero
   D. Because mu is always zero
   Answer: B. E[X - mu] = mu - mu(1) = 0 using the mean definition and total area of the PDF equal to 1. (Key point: First central moment is zero)

2. The noise equivalent bandwidth of an RC low-pass filter is
   A. 1/(2 pi RC)
   B. 1/(4RC)
   C. 1/(RC)
   D. pi RC / 2
   Answer: B. Integrating 1/(1+(2 pi f RC)^2) from 0 to infinity gives (1/(2 pi RC))(pi/2) = 1/(4RC). (Key point: NEB of RC low-pass filter)

3. How does the NEB of the RC-LPF relate to its 3 dB bandwidth?
   A. B_N = f_3dB
   B. B_N = (pi/2) f_3dB, about 1.57 f_3dB
   C. B_N = f_3dB / 2
   D. B_N = 2 f_3dB
   Answer: B. With f_3dB = 1/(2 pi RC), B_N = 1/(4RC) = (pi/2) f_3dB. (Key point: NEB versus 3 dB bandwidth)

4. White noise of two-sided PSD N_0/2 passes through the RC-LPF. The output noise power is
   A. N_0/(2RC)
   B. N_0/(4RC)
   C. N_0 RC/4
   D. N_0/(8RC)
   Answer: B. P_out = (N_0/2)(2 B_N) = N_0 B_N = N_0/(4RC). (Key point: Output noise power of RC-LPF)

## Related in Communication Systems

- [Ideal and RC filtering of white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arc-filtering-white-noise) — Communication Systems, chapter 9
- [Noise equivalent bandwidth of RC low pass filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-rc-lowpass) — Communication Systems, chapter 9
- [Noise equivalent bandwidth; optimum detector impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-optimum-detector) — Communication Systems, chapter 9
- [Noise equivalent bandwidth; matched filter is time reversed](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-matched-filter-reverse) — Communication Systems, chapter 9
- [Random process; white noise PSDF and autocorrelation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-white-noise) — Communication Systems, chapter 9, asked 2×
- [Optimum detector impulse response in additive white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response) — Communication Systems, chapter 9, asked 2×

## Related in other subjects

- [Frequency response; ideal LPF impulse response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-cutoff-impulse) — Signals and Systems, chapter 5, asked 2×
- [Continuous random variable; pdf properties](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acontinuous-rv-pdf-properties) — Probability and Statistics, chapter 3
- [Impulse response; ideal LPF formula](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-response-ideal-lpf) — Signals and Systems, chapter 5, asked 2×
- [Ideal LPF impulse response; realizable?](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-realizable) — Signals and Systems, chapter 5