Web Site Performance and Signal

The performance of any web site can be analyzed in the Fourier sense during a given time-window provided comprehensive statistics exist for that time-window.

Before we attempt an analysis, we need to decide what exactly constitutes a signal for a web site. The most prominent characteristic of any web site is the number of pageloads (L) or hits it receives, therefore we can define the signal of a web site to be related to the number of hits the web site receives.

Owners of web sites are primarily interested in increasing their web site's hits because this increases the web site's popularity, therefore if the hits go up the performance is becoming better and if the hits go down the performance is worsening.

We are therefore interested in the total variance of hits, or, in other words, in tracking the ups and downs of hits for the web site in question. If p is the period of the time-window in question, we have available data about the total number of hits/p as L[n] from a web page's statistics counter, therefore we can define the web site variance, as:

web site variance (1)

In turn, (1) now defines a web site signal in terms of the Heaviside step function H, as:

web site signal (2)

Once we have a signal S(t), we can analyze it in the Fourier sense using Maple. Follows the analysis for this web site starting with year 2011[1].

Weekly Performance

Week # Hits[2] S(t)/F25(t)[3] Spectrum[4] Trend[5]
390,539,434,449,515,479,345 week 1 web site signal/Fourier week 1 web site spectrum -
438,452,552,484,524,575,505 week 2 web site signal/Fourier week 2 web site spectrum +
532,498,576,543,611,608,604 week 3 web site signal/Fourier week 3 web site spectrum +
480,540,539,577,472,457,343 week 4 web site signal/Fourier week 4 web site spectrum -
442,538,648,477,448,473,397 week 5 web site signal/Fourier week 5 web site spectrum -
340,479,481,608,684,527,421 week 6 web site signal/Fourier week 6 web site spectrum +
468,593,650,564,592,496,441 week 7 web site signal/Fourier week 7 web site spectrum -
447,686,636,707,521,498,472 week 8 web site signal/Fourier week 8 web site spectrum +
518,571,563,627,585,519,464 week 9 web site signal/Fourier week 9 web site spectrum -
578,607,608,656,640,539,429 week 10 web site signal/Fourier week 10 web site spectrum -

Monthly Performance

Month Hits[2] S(t)/F25(t)[3] Spectrum[4] Trend[5]
January 3199,3624,3920,3370 January web site signal/Fourier January web site spectrum +
February 3321,3625,3804,3967 February web site signal/Fourier February web site spectrum +
March 3847,4057,3827,3825 March web site signal/Fourier March web site spectrum -
April 4270,4005,3987,3494 April web site signal/Fourier Apr web site spectrum -

Yearly Performance

Year Hits[2] S(t)/F25(t)[3] Spectrum[4] Trend[5]
2011 15260,14006,16457,14293,13414,10568,
2011 web site signal/Fourier 2011 web site spectrum +


  1. Starting with year 2011, mainly because StatCounter has been unreliable as a data collection source in the past because of power outages which caused loss of data. Even disregarding these previous problems, the StatCounter servers have lately been blocked in many US locations (many standard "hosts" files distributed on the net now include blocks to StatCounter), and the blocked locations continue to increase as a function of time, so the data is now completely unreliable as it misses a large percentage of visitors.
  2. Daily, weekly or monthly web site hits for time-window in question as recorded on StatScounter for the first six weeks, then as recorded on Google Analytics for the indicated dates. Hits by the author to his own web site have been excluded.
  3. Weekly, monthly or yearly signal S(t) (2) for this web site (red) and Fourier Series approximation with 25 terms (green).
  4. Weekly, monthly or yearly Harmonic Spectrum of S(t), showing amplitude of first 10 harmonics. From left to right: 0-th harmonic (DC-term) (red), 1-st harmonic (dominant) (green), 2-nd harmonic (yellow), 3-rd harmonic (blue), 4-th harmonic (magenta), 5-th harmonic (cyan), etc. Note that the n*p-th harmonic for n \in N is always 0. Note sharp change in intensity of harmonics during February, the month that the switch from StatCounter to Google Analytics took place.
  5. vp>0, Trend: +. vp<0, Trend: -. vp=0, Trend: undefined.