Login
OR
Don't worry. We'll fix this for you!
In electronics and telecommunication engineering, signals often carry useful information, but they are also accompanied by unwanted disturbances called noise. Noise is a random signal that can obscure or distort the original message. Understanding noise is essential because it limits the performance of communication systems, sensors, and electronic devices.
Among various types of noise, white noise holds a special place due to its simple and idealized properties. White noise is a random signal with equal intensity at different frequencies, much like white light contains all colors equally. This makes white noise a fundamental concept in signal processing, system design, and performance analysis.
By studying the properties of white noise, engineers can predict how noise affects signals, design filters to reduce noise, and simulate realistic scenarios for testing systems. This section will build your understanding of white noise from the ground up, starting with what it is, how it behaves statistically, and why it matters in practical applications.
White noise is a type of random signal characterized by having a constant power spectral density (PSD) across all frequencies. This means that white noise contains equal power at every frequency within its bandwidth, making it a "flat" noise spectrum.
To understand this better, let's first recall what a random signal is. A random signal is one whose values cannot be predicted precisely and vary in an unpredictable manner over time. Unlike deterministic signals (such as sine waves), random signals are described statistically.
Key statistical properties of white noise include:
Below is a graphical illustration showing a sample white noise waveform and its flat power spectral density.
The zero mean property means that the noise fluctuates symmetrically around zero. This is important because it ensures that the noise does not introduce any bias or offset to the signal it contaminates.
The variance \( \sigma^2 \) quantifies the average power of the noise. A higher variance means stronger noise, which can degrade signal quality more severely.
The autocorrelation function measures how similar a signal is to a time-shifted version of itself. For a random signal \( n(t) \), the autocorrelation at lag \( \tau \) is defined as:
\[ R_n(\tau) = E[n(t) \cdot n(t + \tau)] \]
where \( E[\cdot] \) denotes the expectation (average) over time or ensemble.
For white noise, samples at different times are uncorrelated, meaning the autocorrelation is zero for all non-zero lags. At zero lag, the autocorrelation equals the variance \( \sigma^2 \), reflecting the power of the noise at the same instant.
This behavior can be expressed mathematically as:
\[ R_n(\tau) = \sigma^2 \delta(\tau) \]
where \( \delta(\tau) \) is the Dirac delta function, an impulse at \( \tau = 0 \).
The autocorrelation function looks like an impulse at zero lag, indicating no correlation between different samples.
Autocorrelation tells us how noise values relate over time. For white noise, the lack of correlation means each noise sample is independent, which simplifies analysis and design of systems affected by noise.
The power spectral density (PSD) describes how the power of a signal is distributed over frequency. It is the Fourier transform of the autocorrelation function.
Since the autocorrelation of white noise is an impulse at zero lag, its Fourier transform is a constant function. This means the PSD of white noise is flat across all frequencies.
Mathematically, the PSD \( S_n(f) \) is given by:
\[ S_n(f) = \frac{\sigma^2}{2} \]
This constant PSD implies that white noise contains equal power at every frequency, similar to how white light contains all colors equally.
Knowing that white noise has a flat PSD helps engineers identify noise characteristics in the frequency domain. It also aids in designing filters to reduce noise by attenuating specific frequency bands.
Step 1: Recall the autocorrelation function for white noise is:
\[ R_n(\tau) = \sigma^2 \delta(\tau) \]
Step 2: Substitute the given variance \( \sigma^2 = 4 \):
\[ R_n(\tau) = 4 \delta(\tau) \]
Step 3: Interpretation: The autocorrelation is zero for all \( \tau eq 0 \), and equals 4 at \( \tau = 0 \). This means noise samples at different times are uncorrelated, and the noise power is 4.
Answer: \( R_n(\tau) = 4 \delta(\tau) \), indicating uncorrelated noise samples with power 4.
Step 1: Use the PSD formula for white noise:
\[ S_n(f) = \frac{\sigma^2}{2} \]
Step 2: Substitute \( \sigma^2 = 2 \):
\[ S_n(f) = \frac{2}{2} = 1 \]
Step 3: The PSD is constant and equals 1 for all frequencies.
Step 4: Plotting \( S_n(f) \) would show a flat horizontal line at 1 across the frequency axis.
Answer: \( S_n(f) = 1 \) (constant), confirming the flat PSD characteristic of white noise.
Step 1: The output PSD is given by:
\[ S_y(f) = S_n(f) \cdot |H(f)|^2 \]
Since input noise is white with variance \( \sigma^2 = 3 \), input PSD is:
\[ S_n(f) = \frac{3}{2} \]
Step 2: Substitute values:
\[ S_y(f) = \frac{3}{2} \cdot \frac{1}{1 + f^2} \]
Step 3: Total output noise power \( P_y \) is the integral of \( S_y(f) \) over all frequencies:
\[ P_y = \int_{-\infty}^{\infty} S_y(f) df = \frac{3}{2} \int_{-\infty}^{\infty} \frac{1}{1 + f^2} df \]
Step 4: The integral \( \int_{-\infty}^{\infty} \frac{1}{1 + f^2} df = \pi \).
Step 5: Calculate total output noise power:
\[ P_y = \frac{3}{2} \times \pi = \frac{3\pi}{2} \approx 4.712 \]
Answer: Output noise PSD is \( S_y(f) = \frac{3}{2(1 + f^2)} \), and total output noise power is approximately 4.712.
Step 1: For Gaussian noise, the probability density function (PDF) is:
\[ p(n) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{n^2}{2\sigma^2}} \]
Step 2: With \( \sigma^2 = 1 \), the PDF simplifies to:
\[ p(n) = \frac{1}{\sqrt{2\pi}} e^{-\frac{n^2}{2}} \]
Step 3: The probability that \( n \) lies between -1 and 1 is the cumulative distribution function (CDF) difference:
\[ P(-1 \leq n \leq 1) = \Phi(1) - \Phi(-1) \]
where \( \Phi(x) \) is the standard normal CDF.
Step 4: Using standard normal tables or calculator:
\( \Phi(1) \approx 0.8413 \), \( \Phi(-1) = 1 - \Phi(1) = 0.1587 \)
Step 5: Calculate probability:
\[ P = 0.8413 - 0.1587 = 0.6826 \]
Answer: Approximately 68.26% of white noise samples lie between -1 and 1.
Step 1: In MATLAB, use the command:
n = randn(1, 1000);
In Python (with NumPy):
import numpy as np
n = np.random.randn(1000)
Step 2: Calculate sample mean and variance:
MATLAB:
mean_n = mean(n);
var_n = var(n);
Python:
mean_n = np.mean(n)
var_n = np.var(n)
Step 3: The sample mean should be close to 0, and variance close to 1, confirming white noise properties.
Answer: Simulation produces white noise samples with approximately zero mean and unit variance.
When to use: When analyzing or calculating autocorrelation functions.
When to use: During spectral analysis or filtering questions.
When to use: Exam preparation and time management.
When to use: When practicing or verifying theoretical concepts.
When to use: While revising or explaining concepts.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|
