In a previous post, we introduced the frequency-smoothing method (FSM) of spectral correlation function (SCF) estimation. The FSM convolves a pulse-like smoothing window with the cyclic periodogram to form an estimate of the SCF. An advantage of the method is that is allows fine control over the spectral resolution of the SCF estimate through the choice of , but the drawbacks are that it requires a Fourier transform as long as the data-record undergoing processing, and the convolution can be expensive. The expense of the convolution can be mitigated by using rectangular .
In this post, we introduce the time-smoothing method (TSM) of SCF estimation. Instead of averaging (smoothing) the cyclic periodogram over spectral frequency, multiple cyclic periodograms are averaged over time. When the non-conjugate cycle frequency of zero is used, this method produces an estimate of the power spectral density, and is essentially the Bartlett spectrum estimation method. The TSM can be found in My Papers  (Eq. (54)), and other places in the literature.
The basic idea is to segment the provided data record into contiguous blocks of samples each, compute the cyclic periodogram for each block, and average the results. Since we will likely use the FFT to compute the Fourier transform, we will be viewing each -sample block as if its time samples correspond to , and so the cyclic polyspectrum formula of My Papers  will have to be slightly modified to take into account the actual temporal start time for each block.
So let’s consider the Fourier transform (DFT) of a block of data that is shifted from the origin by some amount of time ,
The periodogram and cyclic periodogram are then functions of time offset as well,
and similarly for the conjugate cyclic periodogram. The TSM estimate of the SCF is simply the average value of the cyclic periodogram over all available values of ,
where is some pulse-like temporal window. In practice, the FFT is used to create each cyclic periodogram, so their relative phases are no longer taken into account. According to our Fourier transform result for a delayed signal, however, we can easily take this into account by multiplying each cyclic periodogram by , where represents the left edge (starting point) of the subblock. For blocks having length samples, then, the value of for the th block is simply . Our TSM estimator is then
where is just the cyclic periodogram created from the th block of samples using the FFT. Notice that when the cycle frequency is set to zero, the SCF estimate is an estimate of the PSD, and the TSM just averages periodograms, as in the Bartlett spectrum estimation method. Here is the TSM (Bartlett) PSD estimate for our rectangular-pulse BPSK signal:
For this PSD estimate the data-record length is samples and the TSM block length is samples, leading to blocks. Recall that the bit rate for the BPSK signal is and the carrier frequency is (in normalized frequency units).
The TSM PSD estimate matches the FSM PSD estimate in the FSM post.
The TSM-based spectral correlation function estimates for the BPSK signal’s non-conjugate cycle frequencies are shown below:
and the conjugate-SCF estimates are:
Again, these TSM estimates match quite well with the FSM estimates.
The reason the TSM and FSM estimates match so well is that the temporal and spectral resolution parameters of the estimates are similar. For both methods, the temporal resolution is equal to the data-record length ( samples). For the FSM, the spectral resolution of the estimates is equal to the width of the frequency-smoothing window , and for the TSM, the spectral resolution is equal to the intrinsic spectral resolution of each cyclic periodogram, which is equal to the reciprocal of the TSM block length (in normalized units).
For the FSM results in the FSM post, the spectral resolution is Hz points in , and for the TSM results in this post, the spectral resolution is Hz. So the two estimates have comparable time and frequency resolution parameters, and so produce similar results. The relationship between estimator quality and the temporal, spectral, and cycle-frequency resolutions is discussed in this post.