High Frequency Brain Signals (HFBS) in EEG and MEG, Part One: A delusion from spectral leakage?

Update 10/23/2013: If the sampling frequency is 2000Hz, the highest frequency that windowing function can introduce is 1000Hz. If 1000Hz signal is observed in digital signal, it does not necessarily mean that the analog signal has 1000Hz component. In this case, the 1000Hz signal may not be from the brain but the sampling technique used.

About a month ago, I met a friend in Cincinnati Children's Hospital on my way from Texas to Ohio. His research is about high frequency brain signals (HFBS, >70Hz) in MEG. He told me that many reviewers couldn't quite understand why there are signals up to hundreds of Hertz in MEG because neuronal activities cannot be that fast.

As a man of science, I strongly believe in my friend's hypothesis but at the same time remain equally skeptical.

Hence, I decide to write 2 blog posts to first challenge him and then support him. Today, I am gonna give him some hard time: it is scientifically reasonable that HFBS may be a delusion caused by a phenomenon known as spectral leakage. 

Short version: One can get high amplitude at any frequency (between 0Hz and half of the sampling frequency) from even a constant signal (i.e., exactly 0Hz) - if you use a bad window function when sampling the data.

Here is the detail. Today, almost all neurophysiological signals, including EEG and MEG, are sampled using Digital Signal Processing (DSP). In order to digitize data from the analog world, we sample them by applying a window function. A bad window function, e.g., a rectangular window, can give you high amplitude from 0Hz to half of the sampling frequency in spectral domain, like this:

Rectangular window and its Fourier Transform. Leaked frequencies are relatively high.
Source: Wikipedia (You will see the same in any DSP textbook.)

This phenomenon is called spectral leakage. You can consider it as frequency components not in the signal "leak" into the spectrum.

Using a better window function, such as Hamming window, can reduce the effect of spectral leakage. For example, the amplitude(s) of "real" frequency(-ies) can be much higher than those of leaked frequencies.

Rectangular window and its Fourier Transform. Leaked frequencies are 1000 times (-60dB) lower than the main frequency. Source: Wikipedia (You will see the same in any DSP textbook.)

But sadly we cannot eliminate spectral leakage. Hence, you always have leaked frequencies while their amplitudes may not be small. The question is, are the amplitudes of HFBS high enough? I need to read more papers on this topic before I can make a claim.

If you wanna understand mathematically why we can never get rid of spectral leakage, go grab any DSP textbook (unfortunately, lots of math) and you will get it.

Comments are welcome, especially those from DSP or neuroscience field.

References:

• Xiang et al., Neuromagnetic correlates of developmental changes in
  endogenous high-frequency brain oscillations in children: A wavelet-based beamformer study, Brain Research, 1274 (2009):28--39
  [The discussion part of this paper explains the sources of low frequency neuronal activities and high frequency ones. ]
• Xiang et al., Noninvasive localization of epileptogenic zones with ictal high-frequency neuromagnetic signals, Journal of Neurosurgery: Pediatrics,
  2010 Jan; 5(1):113--22.