Ferkans — Interactive Telecom Tutor

Why White Noise?

In every communication receiver there is thermal noise — electrons jiggling at random in the front-end amplifier. Over the bandwidth of interest, the spectrum of this noise is essentially flat. The simplest mathematical model that captures "flat spectrum" is white noise: a process whose PSD is constant for all frequencies. This idealization is extraordinarily useful, but it comes with a price — infinite total power — that we must handle carefully.

Definition:
White Noise (Continuous-Time)

A WSS process $W(t)$ is called white noise with (two-sided) spectral density $N_0/2$ if

${P_x}_{w}(f) = \frac{N_0}{2}, \qquad \forall\, f \in \mathbb{R}.$

Equivalently, the autocorrelation is

${r_{xx}}_{w}(\tau) = \frac{N_0}{2}\,\delta(\tau).$

The parameter $N_0$ has units of W/Hz and is called the one-sided noise spectral density.

The name "white" comes from the analogy with white light, which contains all visible frequencies in roughly equal proportion.

,

White Noise Has Infinite Power

The total power of white noise is $\int_{-\infty}^{\infty} \frac{N_0}{2}\,df = \infty$ . This means white noise is a mathematical idealization — no physical process has a truly flat spectrum out to infinite frequency. The model is nevertheless valid whenever the noise bandwidth far exceeds the signal bandwidth, which is the typical situation in communications.

Definition:
Band-Limited White Noise

A WSS process $W(t)$ is band-limited white noise with bandwidth $W$ and spectral level $N_0/2$ if

${P_x}_{w}(f) = \begin{cases} N_0/2, & |f| \leq W/2, \\ 0, & |f| > W/2. \end{cases}$

The autocorrelation is

${r_{xx}}_{w}(\tau) = \frac{N_0\,W}{2}\,\operatorname{sinc}(W\,\tau),$

where $\operatorname{sinc}(x) = \sin(\pi x)/(\pi x)$ . The total power is $\sigma^2 = N_0\,W/2$ .

Example: Decorrelation Time of Band-Limited White Noise

Band-limited white noise has bandwidth $W = 10$ MHz. Find the first zero of the autocorrelation and interpret it.

Solution

Find the zero

${r_{xx}}_{w}(\tau) = 0$ when $\operatorname{sinc}(W\,\tau) = 0$ , i.e., $W\,\tau = \pm 1, \pm 2, \ldots$ . The first zero is at $\tau_0 = 1/W = 100$ ns.

Interpretation

Samples taken more than $100$ ns apart are approximately uncorrelated. This is the reciprocal bandwidth rule: the decorrelation time is $\sim 1/W$ . A wider band means faster decorrelation.

White Noise and Band-Limited White Noise

Compare the PSD and autocorrelation of ideal white noise versus band-limited white noise. Observe how limiting the bandwidth turns the $\delta(\tau)$ autocorrelation into a sinc function.

Parameters

W

(bandwidth, MHz)10

N_0

(spectral level)1

Show ideal white noise

From White to Colored: Bandwidth and Autocorrelation

See how increasing bandwidth flattens the PSD toward the white noise limit while the autocorrelation narrows toward

\delta(\tau)

.

As

W \to \infty

, the sinc autocorrelation narrows and the PSD approaches the constant

N_0/2

.

Definition:
White Noise (Discrete-Time)

A WSS sequence $W_n$ is discrete-time white noise with variance $\sigma^2$ if

${r_{xx}}_{w}[m] = \sigma^2\,\delta[m], \qquad {P_x}_{w}(f) = \sigma^2, \quad f \in [-\tfrac{1}{2}, \tfrac{1}{2}].$

Unlike the continuous-time case, the DT white noise process has finite power: $\mathcal{P}_w = \int_{-1/2}^{1/2} \sigma^2\,df = \sigma^2$ .

Example: Filtering White Noise Produces Colored Noise

Discrete-time white noise $W_n$ with $\sigma^2 = 1$ is the input to an LTI system with impulse response $h[n] = a^n u[n]$ , $|a| < 1$ . Find the output PSD ${P_x}_{y}(f)$ .

Solution

Frequency response

$\check{h}(f) = \sum_{n=0}^{\infty} a^n e^{-j2\pi fn} = \frac{1}{1 - a\,e^{-j2\pi f}}$ .

Output PSD

Since the input is white with PSD $\sigma^2 = 1$ ,

${P_x}_{y}(f) = |\check{h}(f)|^2 \cdot 1 = \frac{1}{|1 - a\,e^{-j2\pi f}|^2} = \frac{1}{1 - 2a\cos(2\pi f) + a^2}.$

Interpretation

The output is "colored" noise — its PSD is no longer flat. For $a > 0$ the spectrum peaks at $f = 0$ (low-pass). For $a < 0$ it peaks at $f = 1/2$ (high-pass). The filter shapes the noise spectrum.

Quick Check

Continuous-time white noise with $N_0/2 = 10^{-9}$ W/Hz passes through an ideal bandpass filter of bandwidth $W = 1$ MHz. What is the output noise power?

$10^{-3}$ W

$\infty$

$10^{-9}$ W

$2 \times 10^{-3}$ W

Correction:

10^{-3}

W

$\sigma^2 = \frac{N_0}{2} \cdot W = 10^{-9} \cdot 10^6 = 10^{-3}$ W $= 1$ mW.

Common Mistake: $N_0$ vs. $N_0/2$ : One-Sided vs. Two-Sided

Mistake:

Writing ${P_x}_{w}(f) = N_0$ (one-sided level) and then integrating over $(-\infty, \infty)$ , getting twice the correct power.

Correction:

Convention: $N_0$ is the one-sided PSD (defined for $f \geq 0$ ). The two-sided PSD is $N_0/2$ . When integrating over all frequencies (positive and negative), use $N_0/2$ . When integrating over $f \geq 0$ only, use $N_0$ . Always state which convention you are using.

🔧Engineering Note

Thermal Noise in Communication Receivers

The one-sided thermal noise spectral density at temperature $T$ (Kelvin) is $N_0 = k_B T$ , where $k_B = 1.38 \times 10^{-23}$ J/K is Boltzmann's constant. At room temperature ( $T = 290$ K), $N_0 \approx 4 \times 10^{-21}$ W/Hz, or equivalently $-174$ dBm/Hz. This sets the fundamental noise floor for any communication receiver. In practice, the noise figure of the receiver chain raises the effective noise temperature above $290$ K.

Practical Constraints

•
Valid for $hf \ll k_B T$ (Rayleigh-Jeans regime), i.e., $f \ll 6$ THz at room temperature

Why This Matters: The Noise Floor in Wireless Systems

Every wireless link budget begins with the noise floor $N_0\,W$ , the total noise power in the receiver bandwidth. The ratio of received signal power to this noise power is the SNR, which determines the achievable data rate via Shannon's capacity formula. Understanding PSD is essential because the noise is not always white over the band of interest — interference, co-channel users, and filtering all shape the effective noise spectrum.

Historical Note: Johnson-Nyquist Noise

1928

John B. Johnson experimentally measured thermal noise in resistors at Bell Labs in 1926--1928. Harry Nyquist provided the theoretical explanation using thermodynamic arguments, deriving the famous formula ${P_x}_{w}(f) = k_B T$ (one-sided). This was one of the first rigorous connections between statistical mechanics and electrical engineering, and it established that thermal noise is fundamentally white over the frequencies relevant to electronics and communications.

White Noise: CT vs. DT vs. Band-Limited

Property	CT White Noise	DT White Noise	Band-Limited White
$P_x(f)$	$N_0/2$ (all $f$ )	$\sigma^2$ ( $\|f\| \leq 1/2$ )	$N_0/2$ ( $\|f\| \leq W/2$ )
Autocorrelation	$(N_0/2)\,\delta(\tau)$	$\sigma^2\,\delta[m]$	$(N_0W/2)\,\operatorname{sinc}(W\tau)$
Total power	$\infty$	$\sigma^2$	$N_0W/2$
Physical?	Idealization	Realizable	Realizable
Decorrelation	Instantaneous	One sample	$\sim 1/W$

White noise

A WSS process with flat PSD: $P_x(f) = \text{const}$ . Continuous-time white noise has infinite power and serves as an idealization; discrete-time white noise has finite power.

Thermal noise

Noise generated by thermal agitation of charge carriers in a conductor. Has one-sided PSD $N_0 = k_B T$ at temperature $T$ . Also called Johnson-Nyquist noise.

Key Takeaway

White noise is the frequency-domain analog of "completely uncorrelated": flat PSD means $\delta$ -function autocorrelation. It is the universal noise model in communications because thermal noise is approximately white over any practical bandwidth. The key engineering parameter is $N_0$ , the one-sided spectral density, which sets the noise floor.

White Noise

Why White Noise?

Definition: White Noise (Continuous-Time)

White Noise Has Infinite Power

Definition: Band-Limited White Noise

Example: Decorrelation Time of Band-Limited White Noise

Find the zero

Interpretation

White Noise and Band-Limited White Noise

Parameters

From White to Colored: Bandwidth and Autocorrelation

Definition: White Noise (Discrete-Time)

Example: Filtering White Noise Produces Colored Noise

Frequency response

Output PSD

Interpretation

Quick Check

Common Mistake: N0N_0N0​ vs. N0/2N_0/2N0​/2: One-Sided vs. Two-Sided

Thermal Noise in Communication Receivers

Why This Matters: The Noise Floor in Wireless Systems

Historical Note: Johnson-Nyquist Noise

White Noise: CT vs. DT vs. Band-Limited

White noise

Thermal noise

Key Takeaway

Definition:
White Noise (Continuous-Time)

Definition:
Band-Limited White Noise

Definition:
White Noise (Discrete-Time)

Common Mistake: $N_0$ vs. $N_0/2$ : One-Sided vs. Two-Sided