graphic_eq
Autocorrelation Function &
Adaptive Signal Processing
Theory, Properties, and Applications
In-Depth Research Presentation
functions Introduction to Autocorrelation
info Definition
Measure of similarity between a signal and its timedelayed copy. It reveals periodic patterns obscured
by noise.
calculate Discrete-Time Formula
Rxx(k) = E[x(n)x(n+k)]
Rxx(k) = Σ x(n)x(n+k) (Finite Energy)
check_circle Periodicity Detection
check_circle Signal Similarity
Autocorrelation Function & Calculation
settings Mathematical Properties
sync_alt Even Symmetry
Rxx(τ) = Rxx(-τ)
Autocorrelation is symmetric about the origin.
trending_up Maximum at Zero Lag
Rxx(0) ≥ |Rxx(τ)|
Signal is most similar to itself with no delay.
graphic_eq Wiener-Khinchin Theorem
Sxx(f) = F{Rxx(τ)}
Power Spectral Density is the Fourier Transform of Autocorrelation.
Time-Frequency Duality
hub Autocorrelation in Adaptive Signal Processing
settings_input_component Input Correlation Matrix R
The autocorrelation matrix of the input signal determines the
convergence speed and stability of adaptive algorithms like LMS.
R = E[ x(n) xH(n) ]
compare_arrows Eigenvalue Spread
The ratio of maximum to minimum eigenvalues (χ(R)) affects
convergence.
χ(R) = λmax / λmin
lightbulb High spread → Slow convergence. Adaptive filters must
adapt to the input statistics defined by autocorrelation.
Adaptive Filter Framework
graphic_eq Application: Adaptive Line Enhancement (ALE)
waves Detecting Periodicity in Noise
Autocorrelation amplifies periodic components while averaging out
random noise.
filter_alt ALE Mechanism
White Noise ACF
Sine Wave ACF
Impulse at τ=0, zero elsewhere
Periodic peaks at multiples of period
tips_and_updates Adaptive Line Enhancer
A self-tuning filter that uses the difference in autocorrelation
properties to extract narrowband signals from wideband noise.
input
Noisy Input
arrow_forward
tune
ALE Filter
arrow_forward
check_circle
Clean Signal
De-correlation Filter & Normalized ACF
tune Application: Variable Step Size LMS
speed Adaptive Step Size Control
Use autocorrelation of error to adjust step size μ dynamically for
optimal convergence.
VSS Update Equation
μ(n) = α·μ(n-1) + β·e²(n)
Autocorrelation enhancement:
μ(n) = f(e(n)·e(n-1))
north Large Step
south Small Step
When errors are uncorrelated (far
When errors are correlated (near
from optimum)
optimum)
verified Benefits
Fast convergence initially + Low steady-state error + Robustness
to disturbances
LMS Algorithm Signal Flow
Application: Noise Cancellation & Speech
noise_aware
Enhancement
Autocorrelation-Based Noise Subtraction
auto_fix_high (ANS)
Exploits distinct temporal correlation properties of speech and
noise.
Noise ACF
Speech ACF
Decays rapidly to zero
Persists with periodicity (correlated)
(uncorrelated)
psychology Pole-Preserving Property
Autocorrelation of an all-pole signal preserves the same poles →
Enables feature extraction in noisy conditions.
settings_suggest ANS Method
Estimate noise ACF during non-speech periods → Subtract from
noisy speech ACF → Extract clean features.
Autocorrelation in Speech Analysis