3.1 Continuous-time and discrete-time signals

Continuous-time and discrete-time signals are fundamental concepts in bioengineering. Continuous signals, like ECG and EEG, are defined over a continuous time range. Discrete signals, such as MRI and CT scans, are defined at specific time points.

Understanding the differences between these signal types is crucial for processing biomedical data. Sampling converts continuous signals to discrete ones, while reconstruction does the opposite. The Nyquist-Shannon theorem guides proper sampling to avoid aliasing and information loss.

Continuous-Time and Discrete-Time Signals

Continuous vs discrete-time signals

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• Continuous-time signals defined over a continuous range of time represented as a function of a continuous variable, typically denoted as $x(t)$ can take on any value within the continuous time domain (voltage, pressure)
• Discrete-time signals defined only at discrete time instants represented as a sequence of values, typically denoted as $x[n]$ time instants are usually uniformly spaced, with a sampling period $T_s$ (digital audio, video frames)
• Fundamental differences:
  • Domain: continuous-time signals defined over a continuous domain, while discrete-time signals defined over a discrete domain
  • Representation: continuous-time signals represented as functions, while discrete-time signals represented as sequences
  • Values: continuous-time signals can take on any value within the continuous range, while discrete-time signals can only take on values at discrete time instants

Biomedical signal examples

• Continuous-time signals in biomedical applications:
  • Electrocardiogram (ECG) measures electrical activity of the heart continuously over time
  • Electroencephalogram (EEG) measures electrical activity of the brain continuously over time
  • Blood pressure provides continuous measurement of the pressure exerted by circulating blood on the walls of blood vessels
• Discrete-time signals in biomedical applications:
  • Magnetic Resonance Imaging (MRI) uses discrete spatial measurements of the body's internal structures
  • Computed Tomography (CT) scans produce discrete cross-sectional images of the body
  • Photoplethysmogram (PPG) measures discrete blood volume changes in the microvascular bed of tissue

Signal representation conversion

• Sampling converts a continuous-time signal to a discrete-time signal by measuring the value of the continuous-time signal at discrete, uniformly spaced time instants
  • The time interval between consecutive samples is called the sampling period, denoted as $T_s$
  • The sampling frequency, $f_s$, is the reciprocal of the sampling period: $f_s = 1/T_s$
• Reconstruction converts a discrete-time signal back to a continuous-time signal by interpolating between the discrete-time samples to estimate the original continuous-time signal
  • Common interpolation methods include zero-order hold, linear interpolation, and sinc interpolation

Implications of signal sampling

• Nyquist-Shannon sampling theorem states that to accurately represent a continuous-time signal with a discrete-time signal, the sampling frequency must be at least twice the highest frequency component in the continuous-time signal
  • The minimum sampling frequency required to avoid aliasing is called the Nyquist rate
• Aliasing occurs if the sampling frequency is less than twice the highest frequency component in the continuous-time signal
  • Aliasing is the phenomenon where high-frequency components in the original signal appear as lower-frequency components in the sampled signal, leading to distortion and loss of information
• Quantization maps the continuous amplitude values to a discrete set of values when sampling a continuous-time signal, as the amplitude of the signal must be represented using a finite number of bits
  • Quantization introduces an error called quantization noise, which can affect the accuracy of the sampled signal