It is often natural to describe oscillatory signals like the EEG in the frequency domain. Indeed, this is reflected in the convention of describing clinical EEG recordings in terms of activity within frequency bands (e.g., delta, theta, alpha, and beta). In this section we briefly review key concepts from the mathematical analysis of frequency domain or spectral characteristics of signals. This overview will help the reader to better understand common features of spectrograms encountered in the ICU setting, reviewed in section “Interpretation of Spectrograms.”

Before one can understand how to analyze stochastic signals like the EEG, one first needs to understand the principles underlying the more basic theory of spectral decomposition, which we will review now. Figure 4.3 shows the decomposition of a fairly complex 9-s long single-channel EEG signal into a series of sinusoids of varying phases and amplitudes. The black curve in Fig. 4.3a shows the original signal, consisting initially of a low-amplitude “baseline” period and an epileptiform discharge around t = 4 s, followed by a larger amplitude oscillating pattern at approximately 2 Hz that decays in amplitude while slowing in frequency to approximately 1 Hz by the end of the figure window.

The overlaid red curve is an approximation to the black curve, obtained by adding together a series of 50 sine and cosine waves or “components,” with frequencies ranging from 0 to 35 Hz spaced evenly at intervals of approximately 0.7 Hz. The nine components which make the largest contribution to the approximation, i.e., the components with the largest amplitudes, sorted by frequency, are displayed in Fig. 4.3b.

For the most part, the sum of these sinusoidal components faithfully represents the original black signal. Careful scrutiny reveals that the approximation succeeds by a delicate series of constructive and destructive interferences of peaks and troughs. This remarkable balancing act is accomplished automatically by a mathematical formula known as the discrete Fourier series. In the examples shown in Figs. 4.3 and 4.4, the formula used is called the discrete Fourier transform, or DFT.

The amplitude spectrum for a signal is obtained by taking the geometric mean of the amplitudes of the sine-cosine pair for each frequency and plotting these amplitudes as a function of frequency, as shown in Fig. 4.3c.

For the most part, the red approximation or “reconstruction” is faithful to the original black signal. However, in places, the reconstruction is not accurate. These failures are instructive. Note particularly the epileptiform discharge or “spike” that occurs at approximately t = 4 s. Here, the reconstruction is smoother than the original signal. It fails to reproduce the abrupt rise in voltage and subsequent abrupt decrease that constitute the spike. This is because in this reconstruction we deliberately excluded sinusoids above 35 Hz from the reconstruction. The true bandwidth of the signal is evidently greater than 35 Hz. That is, higher frequency components are required in the Fourier sum to capture the more abrupt transitions or “sharper turns” that make up an epileptiform discharge. In this case, had we included components for all frequencies up to the Nyquist sampling rate (in this case, approximately 100 Hz), the reconstruction would have been essentially perfect.

We further illustrate the important relationship between “sharpness” and the presence of higher frequency components in the amplitude spectrum in Fig. 4.4. In Fig. 4.4a, we show a bell-shaped curve, reminiscent of an EEG “slow wave,” together with the components in a frequency decomposition and the corresponding amplitude spectrum. As in the previous figure, there is an underlying black curve and an overlaid red curve which approximates the black curve as a sum of sinusoids with amplitudes calculated by the formula for the DFT. We see that, for a waveform with a blunted or non-sharp morphology, the amplitude spectrum is relatively narrow. By contrast, Fig. 4.4b shows a narrow, spikelike, bell-shaped curve, reminiscent of an epileptiform discharge. We see that a broader range of sinusoids extending to a much higher range of frequencies is required to faithfully represent a spikelike transient . We will see that this observation, that signal “sharpness” manifests in the frequency domain as a broadening of the amplitude spectrum, is fundamental in interpreting spectrograms in ICU EEG monitoring.

Though we have seen that it is possible to decompose complex signals into simple sinusoidal components, there is something unnatural about this decomposition for signals like the one shown in Fig. 4.3. In particular, the signal appears to change character over course of the 9 s shown. It would be more natural to break signals like this into smaller segments or windows, over which the signal characteristics are approximately constant. In statistical jargon, we desire to break the signal into segments that are statistically stationary.

Figure 4.5 shows an example of another signal that shows marked nonstationarity. This example shows a 3-min-long single-channel EEG signal. Figure 4.5b is the raw signal containing a seizure that begins around t = 20 s and ends around t = 120 s. The panels in Fig. 4.5a show the power spectrum (essentially, the amplitude spectrum squared, except for some smoothing—see next below) calculated from three different windows centered at t = 20, 60, and 130 s. The spectra within these three windows differ markedly, reflecting the evolution of signal characteristics that typify seizure activity.

Figure 4.5c shows the result of repeatedly calculating the spectrum of signals in 2-s windows, using a “sliding window” to obtain a new spectrum every 0.1 s. These spectra collectively form an image , or time-frequency spectrogram, formed by representing the power spectrum on a colormap. In this example, the power is shown in decibel (log) units, according to convention. This display convention allows simultaneous visualization of the signal power over a wide range of frequencies in one image, despite the fact that in human EEG the power at different physiologically relevant frequencies can vary by an order of magnitude or more. Representing this EEG signal as a spectrogram clearly brings out the salient dynamical features of the seizure, namely, an increase in the dominant frequency of oscillatory activity and a sharpening of the signal contour, followed by slowing down as the seizure ends. These features are not visible in the raw signal at this scale, though they are clearly evident when examining the signal within a more conventional 10-s window used for clinical review of EEG data. By contrast, these dynamics are clearly evident in the spectrogram despite the “large” 3-min window. The ability of spectrograms to display salient features of the EEG at a zoomed out or “compressed” scale is a major reason that spectrograms are useful in ICU EEG. Further examples at more compressed scales (2 h) will be discussed in subsequent sections.

The principles of spectral analysis described above are exploited to interpret patterns that occur commonly in spectrograms from cEEG recordings in the ICU setting. We illustrate these patterns using synthetic data in Fig. 4.5, before turning to real examples. Figure 4.6a shows a simple signal, a “monotonous” or unchanging low-amplitude sinusoid of 2 Hz. The corresponding spectrogram has a single peak at 2 Hz within every time window that manifests in the spectrogram as a red line at 2 Hz. This signal is reminiscent of the common ICU EEG pattern of “delta slowing” seen in patients with encephalopathy. Figure 4.6b shows a sinusoidal signal with a frequency that begins at 2 Hz, then increases following a linear ramp to 5 Hz while increasing also in amplitude, and then drops abruptly back to 2 Hz. The evolving portion of this pattern is manifest in the spectrogram as an upsloping line. This example is reminiscent of the classic pattern of evolution characteristic of many epileptic seizures and typifies what we will refer to in a subsequent section as a “flame”-type seizure.

Figure 4.6c shows a more complex synthetic signal , consisting of a series of sharp discharges of randomly varying amplitudes, occurring in a regular or periodic fashion at 1 Hz. The spectrogram in this case shows high power not only at 1 Hz, reflecting the periodicity of the repeating pattern, but also a broadening of the spectrum up to approximately 5 Hz and beyond. This broadband character of the spectrogram reflects the fact that the morphology of the discharges is sharp and thus has a broad amplitude or power spectrum, as discussed previously in connection with Fig. 4.4b. Figure 4.6d shows another more extreme example, a sawtooth wave.

In both Fig. 4.6c, d, the pattern can be characterized as broadband monotonous, referring to the repetitive periodic nature combined with the relatively broadband of high power due to the sharpness of the underlying discharges. As will be seen in the real examples below, the broadband-monotonous pattern typifies the spectrogram when the underlying EEG is in a state of either periodic epileptiform discharges or certain closely related states of status epilepticus.

Our discussion regarding spectral estimation has glossed over many important technical details that are critical in certain applications of spectral estimation. We touch briefly on two important fundamental issues that affect the quality of spectrograms.

Consider again the example of a seizure and its spectrogram shown in Fig. 4.5. In that example we chose to make the window size 2 s. This choice in turn dictated a limit to the level of detail with which we were able to resolve temporal features in the EEG, so that the spectrogram has a certain degree of blurring or smoothness across time. If we attempt to obtain higher temporal resolution by making the analysis windows progressively smaller, we would at the same time see that we progressively lose the ability to distinguish detail in the frequency domain. This is because decreasing the signal segment length (keeping the sampling rate constant) reduces the number of frequency components and increases the spacing between components in the DFT. In effect, nearby peaks in the spectrum become single peaks, in the same way that creating a histogram using wide bins can smooth together and obscure peaks in a distribution that has multiple modes.

This consideration highlights a fundamental trade-off that exists between the maximal spectral resolution (the level of detail with which we can calculate the spectrum, i.e., the spacing between frequency components) and the maximal temporal resolution (the smallest window over which we can perform spectral analysis on a signal). The mathematical reasons behind the trade-off between temporal and spectral resolution are in fact identical to those that describe the well-known “Heisenberg uncertainty principle ,” which describes the inverse relationship between the precision with which one can simultaneously measure the position and velocity of a particle.

A second fundamental trade-off arises from the fact that the EEG, like many other naturally occurring signals, is best regarded as stochastic, containing an underlying signal of interest that is corrupted by noise. Consequently, the spectrum and hence the spectrogram need to be estimated. This fact gives rise to a trade-off between bias and variance.

The bias-variance trade-off in spectral estimation is illustrated in Fig. 4.7. In Fig. 4.7a, we show a sample of a stochastic signal generated by a model with a known power spectrum, shown as a red curve in Fig. 4.7b–e. This spectrum has two peaks, at approximately 11 and 14 Hz. Let us assume that we have already chosen a window size for our spectral estimates, based on either the maximum window length over which the signal can be considered to be statistically stationary or the desired level of temporal resolution. In this case we have chosen a window size of 15 s.

Given the chosen window length, a common but suboptimal way to obtain a spectral “estimate” from this finite-length signal is simply to calculate the discrete Fourier transform, usually using a fast algorithm known as the “fast Fourier transform” (FFT), and to then take the squared amplitude of the result as an estimate of the power spectrum. The result is shown as the blue line in Fig. 4.7b. The spectrum appears very noisy. This is not surprising when one considers that what we have done amounts to “estimating” a quantity using only a single sample. Estimates obtained in this way are sometimes called “periodograms.” (We have passed over the fact that, technically, to calculate the periodogram , one usually multiplies the signal by a windowing function or “data taper” before computing the FFT to reduce an effect known as spectral leakage. All spectral estimates shown in this chapter have been computed with appropriate data tapers.)

Under the right conditions in a laboratory, it might be possible to obtain numerous repeated sample segments of the same signal. Given enough repeated samples, we could obtain an accurate estimate of the underlying true spectrum by averaging periodograms. However, in ICU EEG monitoring, we of course do not have the luxury of holding the patient’s state constant to obtain repeated samples. We are thus forced to resort to methods of reducing the variance of the spectral estimates, i.e., of making the spectrogram smoother. Figure 4.7c–e show progressively more aggressive smoothing of the original signal. Smoothing necessarily blurs together fine spectral details, as evidenced by the fact that beyond a certain point the spectral peaks in this example become indistinguishable (Fig. 4.7e).

We operationally define the spectral resolution of the estimated power spectrum as the minimum distinguishable difference between two narrow peaks that can be distinguished in the estimated spectrum. Judging by eye, the approximate spectral resolution in Fig. 4.7c–e is <1 Hz, 2 Hz, and 5 Hz, respectively. The optimal trade-off between variance reduction (smoothness) and spectral resolution in this example appears to be most closely achieved in Fig. 4.6d. Note that the spectral resolution is thus usually lower than the maximal spectral resolution discussed in the preceding subjection. The maximal spectral resolution depended on the window width rather than on statistical considerations.

The most appropriate degree of spectral smoothing clearly depends on the spectral characteristics and intrinsic smoothness of the underlying processes which generate the signal and thus varies depending on the application. In clinical ICU EEG monitoring, a spectral resolution of approximately 0.75 Hz is usually adequate and allows for sufficient spectral smoothing to obtain high-quality spectral estimates. This is the resolution used for the spectrogram in Fig. 4.5 and in the examples shown later.

Numerous approaches to spectral smoothing have been proposed. Common methods include averaging spectra from consecutive neighboring windows (“weighted overlapping segment averaging ,” WOSA) or replacing the amplitudes in a “noisy” spectrum obtained from an appropriately computed discrete Fourier transform by locally weighted averages of neighboring values. Various window functions or “kernels” can be used for this “kernel smoothing” method. The current state of the art, however, is the method called multitaper spectral estimation algorithm (MTSA), which involves averaging together the amplitude spectra of multiple discrete Fourier-transformed segments that have been pre-multiplied by a specific series of windowing functions or “tapers,” known as the discrete prolate spheroidal sequences (DPSS). While the technical details of MTSA are beyond the score of this chapter, it suffices for our purposes to know that MTSA is the solution to a mathematically well-defined optimization problem, designed to achieve a balance between spectral resolution (bias) and the variance of spectral estimates. Surprisingly, though it was invented in the early 1980s, MTSA is not yet in wide use. In fact spectral estimation routines implemented in many commercial products produce relatively poor-quality spectrograms, very often simply “computing the FFT” [18–20].

As with all smoothing methods, MTSA has adjustable parameters that allow one to decide precisely how to balance the bias against variance. In the remaining figures shown in this chapter, spectrograms are computed with a moving window length of 4 s, with overlapping windows shifting by 0.1 s, and a spectral resolution of 0.75 Hz.