# Relation between autocorrelation function and periodogram in time series analysis

I was wondering if anyone could give me some insight on the relation between the ACF and the periodogram of a time series.

I have a bunch of timeseries and their ACF's and periodograms are typically much like the examples below.

For my analysis, I'm mostly interested in periodicity at lag 8 and 16 (for theoretical reasons)

The frequencies 'B' and 'HB' correspond to lag 16 and lag 8 respectively. The time series actually concerns interresponse intervals in musical performance of a piece that consists solely of eighth notes (16 of them in a 4:4 bar so 'B' stand for bar and 'HB' for half bar).

The thing I actually wanted to ask: in my periodograms, I consistently get very large peaks at frequency 0.25 (which corresponds to lag 4). However, the ACF peak at lag 4 is much smaller than those at lag 8 or 16. I was wondering how to interpret this finding. A lot of time series variance can be explained at this frequency even though the lag 4 autocorrelation is quite low?

I hope I was sufficiently clear in my question. If not, don't hesitate to ask me.

The relation between the autocovariance (or autocorrelation) and the spectral density (for which the periodogram is an estimator) is given by the Fourier transform. The two form a so-called Fourier-transform pair meaning the two are time(or space)-domain vs. frequency-domain representations of the same thing. Specifically, if time series $\{X_t\}$ has autovariance function $\gamma(\tau)$ at time lag $\tau$, then the spectral density is defined by $$f(\nu)=\sum\limits_{\tau=-\infty}^{\infty} \gamma(\tau)e^{-2i\pi\nu\tau}.$$ In words, the spectral density partitions the autocovariance as energy-per-hertz of a signal. For example, if you have a deterministic signal with period $t=12$, then the series lagged with itself (ACF) at lag 12 will be perfectly correlated (autocorrelation=1). Subsequently, all power in the spectral density will be concentrated at frequency $1/t$.