Autocorrelation
Understanding Autocorrelation
Think of autocorrelation as sliding a signal over a copy of itself and measuring how much they overlap at each offset. When the copy is perfectly aligned (zero lag), the overlap is maximum. As you slide the copy forward or backward in time, the overlap decreases, unless the signal contains a repeating pattern, in which case the autocorrelation peaks again at the period of that pattern.
This makes autocorrelation the primary tool for extracting periodicity from noisy signals. A weak radar echo buried 20 dB below the noise floor is invisible on an oscilloscope, but when you correlate the received signal against the known transmitted waveform, the echo produces a sharp correlation peak that rises above the noise.
Rxx(τ) = ∫ x(t) · x(t + τ) dt
Discrete Autocorrelation:
Rxx[k] = Σ x[n] · x[n + k]
Where:
τ (or k) = Time lag (delay)
Rxx(0) = Signal energy (always the maximum value)
Wiener-Khinchin Theorem:
Sxx(f) = F{ Rxx(τ) }
The power spectral density is the Fourier transform of the autocorrelation function.
Radar Pulse Compression
Autocorrelation is the engine behind pulse compression radar. The transmitter sends a long chirp pulse (10 to 100 microseconds) that sweeps across a wide bandwidth. The receiver passes the returned echo through a matched filter, which mathematically performs autocorrelation between the received signal and a stored copy of the transmitted chirp.
At the correct time delay (corresponding to the target's range), the autocorrelation produces a sharp compressed spike only nanoseconds wide. The processing gain equals the time-bandwidth product of the chirp:
| Parameter | Long Uncompressed Pulse | Chirp + Pulse Compression |
|---|---|---|
| Pulse Width | 10 μs | 10 μs transmitted, ~10 ns compressed |
| Bandwidth | 100 kHz (1/τ) | 100 MHz (chirp sweep) |
| Range Resolution | 1,500 m | 1.5 m |
| Processing Gain | 0 dB | 30 dB (10 μs × 100 MHz = 1,000) |
| Peak Power Required | Very high (must detect at range) | 1,000x lower (same energy, spread over time) |
CDMA Code Synchronization
In CDMA wireless systems, each user transmits with a unique pseudo-random noise (PN) code. These codes are carefully designed so their autocorrelation function has a single sharp peak at zero lag and near-zero sidelobes everywhere else. During initial synchronization, the receiver slides through all possible code phases, computing the autocorrelation at each offset, until it finds the peak that indicates correct alignment. Gold codes and Kasami code families are specifically optimized for these autocorrelation properties.
Frequently Asked Questions
How does autocorrelation enable radar pulse compression?
A radar transmits a long, frequency-modulated chirp pulse (e.g., 10 microseconds) to put enough energy on target for detection. The receiver then correlates the returned echo against a stored copy of the transmitted chirp. At the correct time delay (the target range), the correlation peaks sharply into a compressed pulse only nanoseconds wide. This gives the radar the energy of a long pulse with the range resolution of a short one, a processing gain of typically 20 to 30 dB.
What is the relationship between autocorrelation and power spectral density?
The Wiener-Khinchin theorem states that the autocorrelation function and the power spectral density (PSD) of a signal are a Fourier transform pair. Taking the Fourier transform of the autocorrelation produces the PSD, and taking the inverse Fourier transform of the PSD produces the autocorrelation. This is why spectrum analyzers can estimate the PSD by computing the FFT of the autocorrelation.
Why do CDMA systems depend on autocorrelation properties?
In CDMA, each user is assigned a unique pseudo-random noise (PN) code. These codes are designed so that their autocorrelation function produces a sharp spike at zero lag and near-zero values at all other lags. This allows the receiver to lock onto the correct code phase during synchronization and reject all other users' codes (which appear as noise). Gold codes and Kasami codes are specifically optimized for these properties.