Harmonic Signals and Noise in Time Series

📅 March 2026 ⏱️ ~10 min read 🏷️ Time Series • Spectral Analysis • ARMA

In many problems, the physical system generating the data imposes a fixed period. Mean temperature (e.g., annual seasonality), radio and television signals, certain seismic waves, and cardiograms are typical examples. In such settings, a harmonic signal (possibly contaminated by noise) is often a natural model.

1) Random-Phase Harmonic Model (No Additive Noise)

Consider the model

Y t = Σ j=1 m A j cos(2π f j t + U j)

where $A j$ and $f j$ are constants, and the phases $U j$ are independent random variables distributed as $Uniform(0, 2π)$ .

Intuition: the random phases ensure stationarity in time (no “preferred” time origin), while the frequencies

f j

represent physically imposed periodicities.

2) Case m = 1 (Single Frequency)

Let

Y t = A cos(2π f t + U), U ~ Uniform(0, 2π)

One can show:

E(Y t) = 0 Var(Y t) = A 2 /2

The autocovariance function is

γ(k) = (A 2 /2) cos(2π f k)

Hence the autocorrelation function (ACF) is

ρ(k) = γ(k)/γ(0) = cos(2π f k)

3) Spectral Density: Why It Becomes “Lines”

The spectral density is the Fourier transform of the autocovariance:

S Y (f) = Σ k=-\infty \infty γ(k) e -2πi f k

For a pure sinusoid with random phase, the spectrum consists of spikes at the harmonic frequencies:

S Y (f) = (A 2 /4) [ δ(f - f 0) + δ(f + f 0) ]

Important note: here δ is the Dirac delta (a “spike” distribution in frequency), not the Kronecker delta used for discrete indices.

4) General Case (m Frequencies)

For

Y t = Σ j=1 m A j cos(2π f j t + U j)

the ACF is a weighted sum of cosines:

ρ(k) = ( Σ j=1 m A j 2 cos(2π f j k) ) / ( Σ j=1 m A j 2)

and the spectrum becomes a sum of lines:

S Y (f) = Σ j=1 m (A j 2 /4) [ δ(f - f j) + δ(f + f j) ]

5) Signal + Noise Model (Realistic)

Real observations are rarely perfectly harmonic. A more realistic model is:

Y t = H t + N t

where $H t$ is the harmonic component and $N t$ is a stationary noise process (often modeled using ARMA).

For a single frequency:

Y t = A cos(2π f t + U) + N t

Then

γ Y (k) = (A 2 /2) cos(2π f k) + γ N (k)

and the spectrum decomposes cleanly:

S Y (f) = (A 2 /4)[δ(f-f 0)+δ(f+f 0)] + S N (f)

Interpretation: the harmonic part creates sharp peaks, while the noise produces a smooth background spectrum.

6) Example + ARMA Connection

Consider

Y t = 4.4 cos(2πt/12 + U) + N t

This is a period-12 signal (monthly seasonality) plus noise. An interesting exercise is to approximate this “signal + noise” process using an ARMA(2,2) model: ARMA models can reproduce oscillatory autocorrelation patterns and may act as a practical approximation when you do not explicitly model the sinusoid.

References

Applied Time Series Analysis