📡Bioengineering Signals and Systems Unit 9 – Sampling Theory and A/D Conversion

Fundamentals of Sampling Theory

• Sampling theory deals with the process of converting continuous-time signals into discrete-time signals
• Involves capturing the essential information of a continuous signal at specific time intervals (sampling instants)
• Sampling frequency ($f_s$) represents the number of samples taken per second, measured in Hertz (Hz)
• Nyquist rate defines the minimum sampling frequency required to avoid loss of information during the sampling process
  • Nyquist rate is twice the highest frequency component present in the signal
• Sampling period ($T_s$) is the time interval between two consecutive samples, calculated as $T_s = 1/f_s$
• Ideal sampling is a mathematical abstraction that multiplies the continuous-time signal with a train of impulses spaced by the sampling period
• Practical sampling methods include sample-and-hold circuits and analog-to-digital converters (ADCs)
• Reconstruction of the original signal from its samples is possible using interpolation techniques (sinc interpolation)

Nyquist-Shannon Sampling Theorem

• States that a band-limited signal can be perfectly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component of the signal
• Mathematically expressed as $f_s > 2f_{max}$, where $f_s$ is the sampling frequency and $f_{max}$ is the maximum frequency component in the signal
• Provides a fundamental limit on the minimum sampling rate required to avoid loss of information
• Undersampling occurs when the sampling frequency is less than twice the maximum frequency component, leading to aliasing
• Oversampling refers to sampling at a rate higher than the Nyquist rate, which can improve signal-to-noise ratio and simplify anti-aliasing filter design
• Band-limited signals have a finite bandwidth, meaning their frequency components are confined within a specific range
• Theorem assumes an ideal low-pass filter for perfect reconstruction, which is not realizable in practice
• Practical reconstruction filters (anti-aliasing filters) have a non-ideal frequency response, resulting in some level of distortion

Aliasing and Anti-Aliasing Techniques

• Aliasing occurs when the sampling frequency is insufficient to capture the highest frequency components of a signal
• Results in high-frequency components being misinterpreted as lower-frequency components in the sampled signal
• Causes distortion and loss of information in the reconstructed signal
• Anti-aliasing techniques are employed to minimize the effects of aliasing
  • Low-pass filtering the signal before sampling to remove frequency components above the Nyquist frequency
  • Increasing the sampling frequency to ensure the Nyquist criterion is met
• Analog anti-aliasing filters are placed before the ADC to limit the signal bandwidth
  • Ideal anti-aliasing filter has a sharp cutoff at the Nyquist frequency, but practical filters have a transition band
• Oversampling can relax the requirements for the anti-aliasing filter by pushing the aliasing components further away from the signal band
• Digital anti-aliasing techniques, such as decimation and interpolation, can be used to reduce the sampling rate while minimizing aliasing
• Aliasing can also occur in digital signal processing when resampling or performing frequency-domain operations

Analog-to-Digital Conversion Process

• Analog-to-digital conversion (ADC) is the process of converting a continuous-time, continuous-amplitude signal into a discrete-time, discrete-amplitude signal
• Involves sampling the analog signal at regular time intervals and quantizing the sampled values into discrete levels
• Sampling is performed by a sample-and-hold circuit, which captures the instantaneous value of the analog signal at each sampling instant
• Quantization assigns each sampled value to one of a finite number of discrete levels, determined by the resolution of the ADC
  • Resolution is typically expressed in bits, with an N-bit ADC having $2^N$ quantization levels
• Encoding converts the quantized values into a digital representation, such as binary or two's complement
• ADC performance is characterized by parameters such as sampling rate, resolution, accuracy, and dynamic range
• Sampling rate determines the maximum frequency component that can be captured without aliasing, according to the Nyquist-Shannon sampling theorem
• ADC architectures include flash, successive approximation, delta-sigma, and dual-slope converters, each with its own trade-offs in terms of speed, resolution, and power consumption
• Anti-aliasing filters are used before the ADC to limit the signal bandwidth and prevent aliasing

Quantization and Resolution

• Quantization is the process of mapping a continuous range of values to a finite set of discrete levels
• Introduces quantization error, which is the difference between the original analog value and its quantized representation
• Quantization error is limited by the resolution of the ADC, with higher resolution resulting in smaller quantization steps and lower quantization noise
• Resolution determines the smallest detectable change in the analog signal, often expressed in bits or as a percentage of the full-scale range
  • An N-bit ADC has $2^N$ quantization levels and can represent $2^N$ distinct values
• Quantization noise is the inherent uncertainty introduced by the quantization process, often modeled as additive white noise
• Signal-to-quantization-noise ratio (SQNR) measures the ratio of the signal power to the quantization noise power, expressed in decibels (dB)
  • SQNR increases by approximately 6 dB for each additional bit of resolution
• Oversampling and noise shaping techniques can be used to improve the effective resolution and SQNR of an ADC
• Dithering intentionally adds a small amount of noise to the input signal before quantization to randomize the quantization error and reduce quantization artifacts
• Non-linear quantization, such as companding (logarithmic quantization), can be used to improve the dynamic range of the ADC for signals with a wide range of amplitudes

Sampling in Biomedical Applications

• Biomedical signals, such as electrocardiogram (ECG), electroencephalogram (EEG), and electromyogram (EMG), are often sampled for digital processing and analysis
• Sampling rate selection depends on the bandwidth of the biomedical signal and the desired temporal resolution
  • ECG signals typically have a bandwidth of 0.05-100 Hz and are sampled at 250-1000 Hz
  • EEG signals have a bandwidth of 0.5-100 Hz and are sampled at 256-1024 Hz
  • EMG signals have a bandwidth of 20-500 Hz and are sampled at 1-10 kHz
• Anti-aliasing filters are crucial in biomedical signal acquisition to prevent high-frequency noise and interference from corrupting the sampled signal
• Resolution requirements vary depending on the application and the dynamic range of the biomedical signal
  • ECG and EEG signals typically require 12-16 bits of resolution
  • EMG signals may require higher resolution (16-24 bits) due to their wider dynamic range
• Wireless biomedical sensors and wearable devices often have limited power and bandwidth, requiring efficient sampling and data compression techniques
• Adaptive sampling techniques can be used to dynamically adjust the sampling rate based on the signal characteristics or the desired level of compression
• Compressed sensing allows for the reconstruction of sparse signals from a reduced number of samples, potentially enabling lower sampling rates and power consumption in biomedical devices

Error Analysis and Signal Recovery

• Sampling and quantization introduce errors in the digitized signal, which can affect the accuracy of subsequent processing and analysis
• Quantization error is the difference between the original analog value and its quantized representation
  • Quantization error is bounded by the quantization step size, which is determined by the ADC resolution
• Sampling error arises from the finite sampling rate and the non-ideal nature of practical sampling systems
  • Sampling error can result in aliasing, jitter, and aperture uncertainty
• Signal-to-noise ratio (SNR) measures the ratio of the signal power to the noise power, including quantization noise and other sources of noise (thermal, flicker, etc.)
• Effective number of bits (ENOB) is a measure of the actual performance of an ADC, considering the effects of noise, distortion, and non-linearity
  • ENOB is calculated from the measured SNR and is often lower than the nominal ADC resolution
• Oversampling and averaging can be used to improve the SNR and ENOB of an ADC by reducing the impact of random noise
• Dithering adds a small amount of noise to the input signal before quantization to randomize the quantization error and reduce quantization artifacts
• Signal recovery techniques aim to reconstruct the original analog signal from its sampled and quantized representation
  • Interpolation methods, such as linear, polynomial, or sinc interpolation, estimate the values between the sampled points
  • Deconvolution techniques can be used to compensate for the non-ideal frequency response of the anti-aliasing filter and improve the signal reconstruction accuracy

Advanced Sampling Techniques

• Non-uniform sampling involves sampling the signal at non-equidistant time intervals, which can be advantageous in certain applications
  • Event-triggered sampling captures samples based on the occurrence of specific events or conditions in the signal
  • Level-crossing sampling takes samples when the signal crosses predefined amplitude thresholds
• Compressive sensing (CS) enables the reconstruction of sparse or compressible signals from a reduced number of samples
  • CS exploits the sparsity of the signal in a transform domain (e.g., Fourier, wavelet) to recover the signal using optimization algorithms
• Bandpass sampling allows for the direct sampling of bandpass signals without the need for analog frequency translation
  • Undersampling the bandpass signal can alias the signal to a lower frequency range, enabling more efficient processing and storage
• Quadrature sampling uses two ADCs with a 90-degree phase shift to sample complex or quadrature signals, such as in-phase and quadrature (I/Q) components
• Time-interleaved ADCs use multiple ADCs in parallel, each sampling the signal at a fraction of the overall sampling rate, to achieve higher effective sampling rates
• Sigma-delta (ΣΔ) modulation combines oversampling and noise shaping to improve the effective resolution and SNR of the ADC
  • ΣΔ modulators use a feedback loop to push the quantization noise to higher frequencies, which can be filtered out during decimation
• Subsampling techniques, such as equivalent time sampling or undersampling, can be used to capture high-frequency signals with lower-speed ADCs by exploiting the periodicity of the signal
• Adaptive sampling adjusts the sampling rate dynamically based on the signal characteristics or the desired level of compression, aiming to optimize the trade-off between sampling rate and signal fidelity