White Noise

White noise is a fundamental model used in communication and signal processing to describe noise whose power is distributed uniformly across all frequencies. Just as white light contains all visible frequencies in equal measure, white noise contains all frequencies with equal power density, making it the most mathematically tractable and commonly used noise model in system analysis.

Understanding white noise is essential for analyzing noise performance of filters, amplifiers, and communication channels. GATE problems on noise figure, SNR, and equivalent noise bandwidth all build directly on the white noise model.

Core Concept Explanation

A random process is called white noise when its power spectral density (PSD) is constant for all frequencies. Mathematically, if S(f) = N0/2 for all f (two-sided), the process is white. The term two-sided means the PSD is defined for both positive and negative frequencies, which is necessary for complex signal analysis using Fourier theory.

Physically, a flat PSD means that equal amounts of noise power exist in every equal-width frequency interval. For example, the noise power in the band 1 to 2 Hz equals the noise power in the band 1001 to 1002 Hz. This is analogous to white light, which contains all colors equally.

A direct consequence of infinite bandwidth is that the total noise power of ideal white noise is infinite, which is physically impossible. White noise is therefore a mathematical idealization. In practice, every real noise source has some cutoff frequency beyond which the PSD drops. If that cutoff is much larger than the system bandwidth, the white noise model is an excellent approximation.

Mathematical Expression

The two-sided PSD of white noise is: Sn(f) = N0/2 for all f. The one-sided PSD (defined for f >= 0 only) is Sn(f) = N0. The autocorrelation function of white noise is the inverse Fourier transform of its PSD.

Since the PSD is constant, the autocorrelation is: Rn(tau) = (N0/2) * delta(tau), where delta(tau) is the Dirac delta function. This means white noise samples at any two different time instants are completely uncorrelated. Even samples separated by an infinitesimally small time difference are statistically independent, which is why white noise has no memory.

When white noise passes through a linear filter with transfer function H(f), the output PSD becomes So(f) = (N0/2)|H(f)|^2. The total output noise power is then the integral of So(f) over all frequencies.

Practical Understanding

In communication receivers, the incoming signal is accompanied by additive white Gaussian noise (AWGN). The term AWGN combines the white noise model (flat PSD) with the Gaussian amplitude distribution. AWGN is the standard channel model in textbooks and GATE problems.

The noise power at the output of a receiver filter is not N0 times the 3 dB bandwidth but N0 times the noise equivalent bandwidth (NEB). The NEB is the bandwidth of a hypothetical ideal rectangular filter that passes the same noise power as the actual filter. For a first-order RC filter, NEB = pi/2 times the 3 dB bandwidth.

Given:
Two-sided PSD N0/2 = 2 x 10^-10 W/Hz, ideal bandpass filter with bandwidth B = 200 kHz

Why this formula applies:
White noise power through a filter equals PSD times noise bandwidth

Formula:
Pn = N0 x B (one-sided convention)

Substitution:
N0 = 2 x (N0/2) = 2 x 2e-10 = 4e-10 W/Hz
Pn = 4e-10 x 200e3

Calculation:
Pn = 4e-10 x 2e5 = 8e-5 W

Final Answer with units:
Pn = 80 microwatts

Exam Tip: For white noise through a filter, always use noise equivalent bandwidth (NEB), not the 3 dB bandwidth. For an ideal rectangular bandpass filter they are equal, but for RC or Butterworth filters, NEB > 3 dB bandwidth.

Key Properties of White Noise

• PSD is constant: Sn(f) = N0/2 (two-sided), meaning equal power in every unit bandwidth interval.
• Autocorrelation: Rn(tau) = (N0/2) delta(tau), confirming that any two samples at different times are uncorrelated.
• Ideal white noise has infinite total power, making it a theoretical model only. Real noise is bandlimited but approximated as white within system bandwidth.
• When white noise passes through a filter, output power = (N0/2) x 2 x NEB = N0 x NEB for one-sided convention.
• Gaussian white noise (AWGN) has Gaussian amplitude distribution in addition to flat PSD and is the standard channel model in GATE problems.

Quick Revision

• White noise: PSD is flat and constant, Sn(f) = N0/2 (two-sided).
• Autocorrelation: Rn(tau) = (N0/2) delta(tau) — samples at different times are completely uncorrelated.
• Total power of ideal white noise is infinite — it is a mathematical model.
• Noise power through filter = N0 x (Noise Equivalent Bandwidth).
• AWGN = white noise + Gaussian amplitude distribution — standard GATE channel model.
• Exam trap: Do not use 3 dB bandwidth directly for noise power calculation unless filter is ideal rectangular.
• Thermal noise is approximately white up to frequencies of several hundred GHz, well beyond most communication system bandwidths.