Random signals and adaptive filtering
Department of Data Communication Networks and Systems
Lecturer Shukhrat Palvanov
What is a Random Signal?
A random signal, also called a stochastic signal or process, is a time-varying quantity whose values cannot be predicted exactly but can be described statistically.�Unlike deterministic signals (such as sinusoids or square waves), random signals exhibit unpredictable variations in amplitude, phase, or frequency over time. Their behavior is governed by probability laws rather than explicit mathematical equations.
Nature of Randomness:�Random signals represent uncertainty or noise present in physical systems such as electrical circuits, wireless channels, or biological processes.
Examples:
Statistical Description of random signals
To understand random signals, we use statistical parameters instead of explicit equations:
Mean value: average level of the signal.
Variance: measure of how much the signal deviates from its mean.
Autocorrelation: measure of similarity between the signal and its time-shifted version.
Power Spectral Density (PSD): frequency domain representation showing power distribution.
Why Random Signals Matter�
Random signals are essential in signal and image processing because:
Random Variables and Random Processes�
In signal and image processing, many signals are uncertain or unpredictable due to noise and other external factors.
�To describe and analyze such signals mathematically, we use the concepts of random variables and random processes.
�These are fundamental tools for modeling and understanding random behavior in signals.
Random Variable
A random variable is a numerical quantity whose value is determined by the outcome of a random experiment.�It provides a way to represent random phenomena mathematically.
Continuous random variable: takes any value within a range (e.g., temperature, voltage noise).
Discrete random variable: takes countable values (e.g., number of errors in a transmission).
A random variable X is characterized by:
Probability density function (PDF): , which shows how probabilities are distributed over possible values.
Cumulative distribution function (CDF): the probability that X does not exceed x.
Example:�If X represents thermal noise voltage, then X may follow a Gaussian (Normal) distribution with mean 0 and variance σ^2.
Random Process�
A random process (or stochastic process) is an extension of the random variable concept to time.�It is defined as a collection of random variables indexed by time:
Each value of t corresponds to one random variable, and the complete set over time forms the random process.
Example:�Thermal noise measured at different times — each sample is random, but together they form a random process X(t).
Sample Function and Ensemble
A sample function (or realization) is one possible waveform of the random process over time.
The ensemble is the set of all possible sample functions.
Each time you observe the process, you may see a different waveform, but all follow the same probability law.
Continuous and Discrete Random Processes
In digital image processing, we mostly deal with discrete random processes, because images and digital signals are sampled and quantized.
Power Spectral Density (PSD)
The Power Spectral Density (PSD) describes how the power of a signal is distributed over frequency.
It is a key tool in analyzing random and noisy signals in both time and frequency domains.
For a wide-sense stationary (WSS) random process X(t)X(t)X(t), the PSD is defined as the Fourier Transform of its autocorrelation function:
White Noise and Colored Noise
White Noise
Colored Noise
Concept of Filtering in Random Signal Processing
Filtering is the process of extracting useful information from a signal while reducing noise or unwanted components.
In random signal processing, filters are used to separate signal and noise, estimate unknown parameters, or enhance system performance.
A filter transforms an input signal x(t) into an output signal y(t):
y(t)=h(t)∗ x(t)
where h(t) is the filter’s impulse response, and * denotes convolution.
Filtering in Random Environments
When input contains random noise, filter design must consider statistical characteristics of both the signal and noise.
Performance is often measured using Mean Square Error (MSE) between the desired signal 𝑑(𝑡) and the output 𝑦(𝑡):
MSE=𝐸[(𝑑(𝑡)−𝑦(𝑡))2]
Adaptive Filtering
Adaptive filtering refers to a class of filtering techniques where the filter parameters change automatically to adapt to variations in the input signal or environment.
Unlike fixed filters, adaptive filters can learn and update their coefficients in real time to minimize the error between the desired and actual outputs.
Why Adaptive Filtering?
Adaptive Filtering
Structure of an Adaptive Filter
Adaptive Filter Algorithms
Adaptive filters rely on learning algorithms to update their coefficients in response to changes in the input data.�These algorithms determine how fast and accurately the filter adapts to varying environments.
General Weight Update Equation
w(n+1)=w(n)+Δw(n)
where the update term Δw(n) depends on the specific adaptation algorithm used.
Main Adaptive Algorithms
1. LMS (Least Mean Squares) Algorithm:
Simplest and most widely used.
Updates weights using instantaneous error:
w(n+1)=w(n)+μx(n)e(n)
μ= step size (controls convergence speed).
Advantages: simple, low computational cost.
Drawbacks: slower convergence for highly correlated inputs.
Main Adaptive Algorithms
2. NLMS (Normalized LMS):
3. RLS (Recursive Least Squares):
4. Kalman Filter:
Applications of Adaptive Filtering
Adaptive filters are widely used in signal, image, and communication systems where the environment or signal statistics change over time.�Their ability to self-adjust makes them ideal for many real-world applications.
1. Noise Cancellation
Removes unwanted noise from signals in real time.
Used in:
Principle: The adaptive filter estimates the noise and subtracts it from the noisy signal.
Applications of Adaptive Filtering
Echo Cancellation
Channel Equalization
Applications of Adaptive Filtering
System Identification
Prediction and Estimation
Image Processing Applications
Tasks for understanding
Task 1 – Random Signal Generation and Analysis�Generate a random signal, analyze its mean, autocorrelation, and power spectral density (PSD).
Task 2 – Adaptive Noise Cancellation Using LMS Filter�Use an adaptive filter to remove noise from a signal.