Convolution vs Correlation
Convolution and correlation are closely related operations that differ by one key step: convolution flips (reflects) one function before sliding, while correlation does not. Mathematically, (f*g)(t) = ∫f(τ)g(t−τ)dτ (convolution flips g), while (f⋆g)(t) = ∫f(τ)g(t+τ)dτ (correlation does not flip). For symmetric functions like Gaussians, the results are identical. Convolution describes system output; correlation measures signal similarity. In the frequency domain, convolution becomes F(ω)·G(ω) while correlation becomes F(ω)·G*(ω) (conjugate).
The Key Difference: To Flip or Not to Flip
Both convolution and correlation involve sliding one function across another and integrating the product at each shift. The critical difference is that convolution reverses (flips) one function before sliding, while correlation uses both functions as-is. Concretely, in convolution (f*g)(t), you evaluate g at the reflected point (t−τ); in correlation (f⋆g)(t), you evaluate g at the unreflected point (t+τ). For even functions where g(t) = g(−t), the two operations give identical results. For asymmetric functions, the results differ — convolution can shift the output in time in a way correlation does not.
Key Formulas
When to Use Convolution vs. Correlation
Use convolution when computing system outputs. If a signal x(t) passes through a system with impulse response h(t), the output is y = x * h. Convolution correctly accounts for causality — the system responds after the input arrives. Use correlation when measuring similarity or detecting known patterns. If you're searching for a specific waveform in a noisy signal, correlate the signal with a template of the waveform you're looking for. The correlation peak tells you where the template matches best. Radar, sonar, GPS, and communications receivers all use correlation for signal detection.
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Open CalculatorFrequency Domain: Conjugate vs. No Conjugate
In the frequency domain, the difference is elegant. Convolution corresponds to F(ω)·G(ω) — plain multiplication. Correlation corresponds to F(ω)·G*(ω) — multiplication with the complex conjugate. The conjugate flips the phase while keeping the magnitude, which is exactly equivalent to the time-domain flip. Autocorrelation — correlating a signal with itself — gives F(ω)·F*(ω) = |F(ω)|², which is the power spectral density. This fundamental relationship connects time-domain correlation to frequency-domain energy distribution.
Cross-Correlation for Time Delay Estimation
One of the most important applications of correlation is finding the time delay between two versions of the same signal. If microphone A records a sound and microphone B records it with a delay of Δt (because the source is closer to A), the cross-correlation of the two recordings peaks at lag Δt. By measuring this lag and knowing the microphone spacing, you can compute the direction of arrival — this is how beamforming and direction-finding systems work. The correlation peak's sharpness depends on the signal bandwidth: wider bandwidth gives sharper peaks and more precise delay estimates.
Autocorrelation: Correlation of a Signal with Itself
The autocorrelation function R_xx(τ) = ∫x(t)x(t+τ)dt measures how similar a signal is to a shifted version of itself. At zero lag (τ=0), autocorrelation equals the signal energy — the maximum possible value. Periodic signals have periodic autocorrelation, with peaks at every multiple of the period. Random noise has autocorrelation that is essentially zero for all nonzero lags — a spike at τ=0 and nothing else. This property makes autocorrelation useful for detecting periodicity hidden in noise and for distinguishing between different types of random processes. The Wiener-Khinchin theorem states that the power spectral density is the Fourier transform of the autocorrelation.
Related Topics in convolution operations
Understanding convolution vs correlation connects to several related concepts: convolution and correlation, and convolution vs cross correlation. Each builds on the mathematical foundations covered in this guide.
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