Spill over of energy into adjacent frequencies is illustrated.
A series is constructed from the following function: input = 10*cos(2*PI*t*f1)+20*sin(2*PI*t*f2)+randomg() where f1 = 26/(t.ct*t.delta) and f2 = 28/(t.ct*t.delta). The periodogram is computed. Then using loops, f1 and f2 are increased by a small amount: f1 = 26.571428/(t.ct*t.delta) and f2 = 28.571428/(t.ct*t.delta). The periodogram now shows leakage of energy from the original two peaks into adjacent frequencies and the two frequencies are not resolved. This is seen on plotpage Periodogram by running Loops. A partial solution is to sample for a longer time to resolve the frequencies, but this may not resolve the leakage problem.
This is a periodogram of an input function with frequencies which are exactly matched by the Fourier transform frequencies (solid line with circles). Then an input function is generated with slightly different frequencies and the energy spills over into adjacent frequencies and the two modified frequencies are not resolved.
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.