3.2 Signal properties: energy, power, and periodicity

Signal properties are crucial in biomedical engineering. Energy and power calculations help us understand signal intensity and distribution over time. These concepts are key for analyzing ECG waveforms, EEG recordings, and other biomedical signals.

Periodic signals, like ECG and respiratory signals, repeat at regular intervals. Understanding their fundamental period is essential for efficient analysis and representation. This knowledge helps in selecting appropriate processing techniques and provides insights into underlying physiological processes.

Signal Properties

Energy and power of signals

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• Energy of a signal represents the total area under the squared magnitude of the signal
  • For a continuous-time signal $x(t)$, calculate the energy using the integral $E = \int_{-\infty}^{\infty} |x(t)|^2 dt$
  • For a discrete-time signal $x[n]$, calculate the energy using the summation $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$
• Power of a signal represents the average energy per unit time
  • For a continuous-time signal $x(t)$, calculate the power using the limit $P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt$
  • For a discrete-time signal $x[n]$, calculate the power using the limit $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$
• Energy and power calculations provide insights into the signal's intensity and distribution over time

Energy vs power signals

• Energy signals have finite total energy but may have infinite power (transient signals)
  • Examples in biomedical contexts include ECG waveforms representing a single heartbeat and evoked potentials like visual evoked potentials (VEP) or auditory evoked potentials (AEP)
  • Typically analyzed using techniques that capture transient or time-limited features (time-domain analysis, wavelet analysis)
• Power signals have finite average power but may have infinite energy (continuous signals)
  • Examples in biomedical contexts include continuous EEG recordings of brain activity and EMG signals representing muscle activity during sustained contractions
  • Often analyzed using techniques that capture spectral or frequency-domain features (Fourier analysis, power spectral density estimation)

Periodic signals and fundamental period

• Periodic signals repeat themselves at regular intervals
  • For a continuous-time signal $x(t)$, it is periodic if $x(t) = x(t + T)$ for all $t$, where $T$ is the fundamental period
  • For a discrete-time signal $x[n]$, it is periodic if $x[n] = x[n + N]$ for all $n$, where $N$ is the fundamental period
• The fundamental period is the smallest positive value of $T$ (continuous-time) or $N$ (discrete-time) for which the periodicity condition holds
  • Examples in biomedical contexts include ECG signals, where the fundamental period corresponds to the duration of a single cardiac cycle, and respiratory signals, where the fundamental period corresponds to the duration of a single breath
• Periodic signals can be efficiently represented using Fourier series or discrete Fourier transform (DFT), allowing for frequency-domain analysis and filtering
• Periodicity can be exploited for signal averaging and noise reduction (averaging multiple ECG cycles to improve signal-to-noise ratio)

Signal properties in biomedical processing

• Understanding signal properties helps in selecting appropriate processing techniques
  • Energy signals are typically analyzed using techniques that capture transient or time-limited features (time-domain analysis, wavelet analysis)
  • Power signals are often analyzed using techniques that capture spectral or frequency-domain features (Fourier analysis, power spectral density estimation)
• Signal properties can provide insights into the underlying physiological processes
  • Changes in ECG periodicity may indicate heart rate variability or arrhythmias
  • Variations in EEG power spectral density may reflect changes in brain states or neurological disorders
• Analyzing signal properties is crucial for accurate interpretation and diagnosis in biomedical applications
  • Identifying abnormalities in ECG or EEG signals based on energy, power, or periodicity changes
  • Monitoring changes in signal properties over time to assess treatment effectiveness or disease progression