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. However, 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. This amounts to a phase compensation of each cyclic periodogram before it enters the averaging operation.
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 final TSM estimator expression is
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), or .
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. The cycle-frequency resolutions for the two estimates are also the same at the reciprocal of the processed data-record length $\Delta\alpha \approx 1/(NM) = 1/32768$. So the two estimates have comparable time, frequency, and cycle-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.
More on the TSM Phase-Compensation Factor
Some readers question the need for the phase-compensation factor in (5), which is given by
In fact it is not needed whenever it is always equal to a constant, such as one. This happens whenever the product is equal to an integer for all values of . A sufficient condition is that the cycle frequency is equal to a harmonic of , that is, . However, the phase factor is needed otherwise. Here are two confirming examples (do your own when you implement the TSM).
First, let’s look at the case where the cycle frequency is just the bit rate of our usual BPSK signal, or . Let’s pick the TSM block length and process a total of samples. We’ll plot the SCF estimate magnitude for the correct TSM and for the TSM without the phase-compensating factor:
In this case, for all integers , and the phase-compensating factor is clearly needed.
On the other hand, consider a BPSK signal with bit rate and carrier . Choose the bit-rate cycle frequency , which is in fact equal to for :
And so in this case the phase-compensating factor is irrelevant since it is equal to one for all the TSM blocks, and the two TSM estimates are identical.