6.2 Time and Frequency Domain Analysis

Time and frequency domain analysis are the core methods for making sense of biomedical signals like ECGs, EEGs, and EMGs. Rather than staring at raw waveforms, these techniques let you quantify what's happening in a signal, whether that means spotting a specific heartbeat event or uncovering a brain rhythm buried in noise.

Time domain analysis excels at identifying specific events and temporal patterns. Frequency domain analysis reveals the underlying oscillations and spectral content. And when a signal's properties shift over time (which most biomedical signals do), time-frequency methods combine both perspectives.

Biomedical Signals in the Time Domain

Time Domain Analysis Techniques

Time domain analysis examines a signal as a function of time, which is the most direct way to look at it. You're working with the signal in the same form it was recorded. Three main types of analysis apply here:

• Amplitude analysis measures signal intensity at different time points. For example, detecting the tall R-peaks in an ECG tells you when each heartbeat occurred and how strong the electrical activity was.
• Duration analysis assesses how long specific events last. The R-R interval in an ECG (time between consecutive R-peaks) directly gives you heart rate. A normal R-R interval at rest is roughly 0.6–1.0 seconds, corresponding to 60–100 beats per minute.
• Morphology analysis examines the shape and structure of waveform features. The QRS complex in an ECG has a characteristic shape, and deviations from that shape can indicate arrhythmias. Similarly, sharp spikes in an EEG may indicate epileptic activity. Morphological features are often the basis for classifying normal vs. abnormal events.

Mathematical and Statistical Techniques

Several statistical measures help quantify time-domain signals:

• Mean gives the average signal value over a time period. For an ECG, you might compute mean heart rate over a recording session.
• Variance ($\sigma^2$) measures how spread out the signal values are around the mean. High variance in R-R intervals, for instance, indicates greater heart rate variability, which is itself a clinically meaningful metric.
• Skewness quantifies asymmetry in the signal's amplitude distribution. A skewness of zero means the distribution is symmetric; positive skewness means the distribution has a longer tail toward higher values. This can characterize the shape of EEG amplitude distributions.
• Kurtosis measures how peaked or flat the distribution is relative to a normal distribution. High kurtosis suggests the presence of sharp outliers or spikes, which makes it useful for detecting transient events like EEG spikes.

These statistics compress an entire signal segment into a single number, which is powerful for comparison and classification but obviously loses temporal detail.

Frequency Domain Analysis of Signals

Frequency Domain Representation

Frequency domain analysis decomposes a signal into its constituent frequencies. Instead of asking "what's the amplitude at time $t$?", you're asking "how much energy exists at frequency $f$?"

The Fourier transform is the mathematical tool that converts a time-domain signal into its frequency-domain representation. For a continuous signal $x(t)$, the Fourier transform is:

$X(f) = \int_{-\infty}^{\infty} x(t) \, e^{-j2\pi ft} \, dt$

In practice, biomedical signals are digitized, so you'll use the discrete Fourier transform (DFT), typically computed via the fast Fourier transform (FFT) algorithm.

Power spectral density (PSD) describes how a signal's power is distributed across frequencies. PSD is especially useful for identifying dominant rhythms. For example, EEG analysis relies heavily on PSD to distinguish frequency bands:

• Delta (0.5–4 Hz): deep sleep
• Theta (4–8 Hz): drowsiness, light sleep
• Alpha (8–13 Hz): relaxed wakefulness, eyes closed
• Beta (13–30 Hz): active thinking, focus

By looking at the PSD, you can determine which bands dominate at any given time, which is far harder to see in the raw time-domain trace.

Spectrogram Analysis

A spectrogram displays frequency content as it changes over time. It's essentially a series of frequency snapshots stacked together, with time on the x-axis, frequency on the y-axis, and color or intensity representing power.

Spectrograms are particularly valuable for non-stationary signals where frequency content shifts. A sleep EEG is a classic example: as a person transitions from wakefulness to deep sleep, the dominant frequencies shift from alpha/beta down to delta. A spectrogram makes these transitions visually obvious.

Other applications include analyzing speech signals and tracking how EMG frequency content changes during sustained muscle contraction.

Time vs Frequency Domain Analysis

Advantages and Limitations

Time domain analysis:

• Provides a direct, intuitive view of the signal's behavior. You can visually identify events like QRS complexes, muscle contractions, or seizure onset.
• Does not inherently reveal frequency content. Two signals can look quite different in time yet share similar spectral properties, or vice versa.

Frequency domain analysis:

• Reveals spectral composition that's invisible in the time domain. A common practical example: 50/60 Hz power line interference in an ECG is immediately obvious in the frequency domain as a sharp spectral peak, even if it's subtle in the time trace.
• Traditional Fourier analysis assumes the signal is stationary, meaning its statistical properties don't change over time. Most biomedical signals violate this assumption to some degree, which limits the usefulness of a single Fourier transform over a long recording.

Interpretation and Efficiency

Time domain representations are generally more intuitive since they match how the signal was actually recorded. Frequency domain analysis requires understanding concepts like spectral leakage, windowing, and resolution, so it has a steeper learning curve.

On the other hand, frequency domain representations can be more efficient for data compression. A signal that requires thousands of time-domain samples might be well-approximated by just a handful of dominant frequency components.

The choice between approaches depends on your question. If you need to measure when an event occurred or how long it lasted, time domain is your tool. If you need to characterize rhythmic activity or identify noise sources, frequency domain is the way to go. For many real biomedical problems, you'll use both.

Time-Frequency Analysis of Signals

Non-Stationary Biomedical Signals

Most biomedical signals are non-stationary: their frequency content changes over time. An EEG shifts between frequency bands as a person moves through sleep stages. An ECG's characteristics change during exercise. An EMG's spectral content shifts as a muscle fatigues.

Standard Fourier analysis gives you the average frequency content over the entire signal, which smears together these changes. Time-frequency analysis methods solve this by tracking how frequency content evolves.

Short-Time Fourier Transform (STFT)

The STFT works by dividing the signal into short, overlapping segments (using a sliding window), then computing the Fourier transform of each segment. The result is a spectrogram showing frequency content at each time point.

The critical trade-off in STFT is window size:

1. A short window gives good time resolution (you can pinpoint when a frequency change occurred) but poor frequency resolution (nearby frequencies blur together).
2. A long window gives good frequency resolution (you can distinguish closely spaced frequencies) but poor time resolution (fast events get smeared in time).

This trade-off is governed by the uncertainty principle: you cannot have arbitrarily fine resolution in both time and frequency simultaneously. You choose your window length based on what matters more for your application. For detecting brief transient events, use a shorter window. For resolving closely spaced frequency components, use a longer one.

Wavelet Transform

The wavelet transform addresses the fixed-resolution limitation of the STFT by using analysis functions (wavelets) that adapt their time-frequency resolution to the scale being analyzed:

• At high frequencies, wavelets are short in duration, providing good time resolution for capturing transient events.
• At low frequencies, wavelets stretch out, providing good frequency resolution for capturing slow oscillations.

This multi-resolution property makes wavelets well-suited for biomedical signals that contain both fast transients and slow trends.

Common wavelet families include:

• Haar wavelets: Simple, square-shaped. Good for detecting abrupt, step-like changes in a signal.
• Daubechies wavelets: Smooth and asymmetric, with good frequency localization. Widely used in general biomedical signal analysis.
• Morlet wavelets: Complex, sinusoidal wavelets. Particularly effective for analyzing oscillatory patterns like EEG rhythms.

The choice of wavelet depends on the signal characteristics and what features you're trying to extract.

Applications of Time-Frequency Analysis

Time-frequency methods are applied across biomedical signal types:

• EEG: Tracking how alpha, theta, and delta power change across sleep stages, or monitoring spectral shifts during cognitive tasks.
• ECG: Analyzing heart rate variability over time and detecting transient arrhythmias or ischemic episodes that cause sudden morphological changes in the QRS complex.
• EMG: Observing how the median frequency of muscle activity shifts downward during sustained contraction, which is a well-established indicator of muscle fatigue.

In each case, time-frequency analysis captures dynamics that neither pure time-domain nor pure frequency-domain methods can reveal on their own.