Sampling and Quantization

Prince E. Adjei

Kwame Nkrumah University of Science and Technology

Topic: Sampling and Quantisation Module 0: Introduction

Biosignal Processes And Analysis (BME 366)

Sampling and Quantization

Topics:

(1). Nyquist-Shannon Theorem and Aliasing

(2). Oversampling and Decimation

(3).Analog-to-Digital Converters(ADC)

(4). Quantization and Its Effects on Signals

(5). Anti-aliasing Filters

Learning Objectives

•Explain the Nyquist-Shannon sampling theorem and its

significance.

•Identify the effects of aliasing and how to prevent it.

•Understand oversampling and decimation in biosignal systems.

•Describe the key components of an analog-to-digital converter

(ADC).

•Evaluate the impact of quantization noise and bit resolution on

signal quality.

•A continuous-time signal is defined for every instant in a continuous

domain, (e.g. x(t)), where time can take any real value. Example ECG.

Review

•A discrete-time signal is defined only at discrete time intervals (e.g.,

x[n]), where time is represented as a sequence of discrete values

(e.g., integers). Example Digital Temperature.

Review

•Sampling:

The process of converting a continuous-time signal into a

discrete-time signal by measuring its value at regular intervals.

•Aliasing:

It occurs when a signal is undersampled. If an ECG signal

containing high-frequency noise is undersampled, the noise may

appear as a lower frequency in the sampled signal, distorting the

signal’s representation.

•A continuous-time signal is defined for every instant in a continuous

domain, (e.g. x(t)), where time can take any real value. Example ECG.

•Many signals are continuous-time in the sense that they are defined at

arbitrarily close points in time.

•A continuous-time sine wave is defined by:

Continuous vs Discrete time signals

where F is the frequency, Ω is the angular frequency, a the amplitude,

and φthe phase.

•A discrete-time signal ( time series) is defined only at discrete time

intervals (e.g., x[n]), where time is represented as a sequence of discrete

values (e.g., integers).

•They are defined over the set of integers, that is, they are indexed

sequences.

•A discrete-time sine wave is defined by :

Continuous vs Discrete time signals

Given an ECG signal:

•Starts as a continuous-time

ECG strip

•Signal is sampled at discrete

time intervals → results in a

sampled series

•Sampled data can be

visualized using a stem plot

to show individual data

points

Nyquist Shannon Theorem

•The Nyquist theorem states that to accurately sample and reconstruct

a continuous signal, the sampling frequency must be at least twice the

signal's highest frequency component.

•This is known as the minimum Nysuist rate. Mathematically:

fs ≥ 2fmax

where: fs is the sampling frequency,

fmax is the highest frequency component of the signal

Aliasing

•If the sampling frequency is

below twice the maximum

frequency of the signal

aliasing occurs.

•It distorts the signal by

introducing high-frequency

components.

•Perfect reconstruction is possible if sampling meets or exceeds the Nyquist

rate

Aliasing

Aliasing in Practice

Oversampling

•Oversampling is sampling a

signal at a rate much higher

than the Nyquist rate (e.g.,

1000 Hz instead of 500 Hz for

a signal with a 250 Hz

bandwidth).

•This captures more data points

to improve signal accuracy and

reduce errors.

1.The Music Telegraph. (n.d.). How to use EQ effectively in your songs.

[1]

Why Oversample in A/D Front-Ends

•Eases Anti-Aliasing Filter Design: Higher sampling rates push aliasing

frequencies further away from the signal’s bandwidth, allowing

simpler, less steep anti-aliasing filters.

•Reduces Quantization Noise: Spreads noise over a wider frequency

range, lowering noise density in the band of interest.

•Improves Resolution: Increases effective bit depth in analog-to-

digital converters (ADCs).

Decimation

•Decimation is the process of reducing the sampling rate (e.g., from

1000 Hz to 250 Hz) after oversampling.

•Involves low-pass filtering to remove high-frequency components

and downsampling to the target rate.

•This produces a lower-rate signal suitable for processing or storage

while retaining quality.

•It improves anti-aliasing and roll-off.

Questions

1.What does the Nyquist theorem

state?

2.What is aliasing and when does it

occur?

3.Why is oversampling followed by

decimation useful in signal

processing?

•Analog-to-digital conversion (ADC) is the process of transforming a

continuous analog signal, such as a biomedical signal, into a discrete

digital signal that a computer can process.

•The conversion process involves:

⚬Sampling

⚬Quantization

⚬Encoding

Analog-to-Digital (ADC) Conversion

•Sampling is the process of measuring the amplitude of an analog

signal at regular intervals to convert it into a discrete-time signal.

•Quantization is the process of mapping the continuous amplitude of

sampled values to a finite set of discrete levels, introducing

quantization noise.

•Encoding is the process of assigning binary codes to the quantized

levels to represent the digital signal in a format suitable for storage

or transmission.

Analog-to-Digital (ADC) Conversion

•The main blocks or parts are: sampler, holding circuit, quantizer and

encoder.

Analog-to-Digital (ADC) Block

[2]

2. Electrical Technology. (2019, February). Analog-to-digital converter (ADC)

Sampler

•The sampler is a circuit that takes samples from the continuous

analog signal according to its sample frequency.

•The sampling frequency is set according to the requirement.

Holder

•The holding circuit does not convert anything it just holds the

samples generated by the sampler circuit.

•It holds the first sample until the next sample comes from the

sampler.

Analog-to-Digital (ADC) Block

Quantizer

•It converts the continuous amplitude-discrete time signal into a

discrete time-discrete amplitude signal.

•It breaks or splits the samples into small parts.

N-bit encoder

•This circuit assigns a binary code (e.g., N bits) to each quantized

level, producing the final digital output.

•The output from the encoder is fed to the next circuitry.

Analog-to-Digital (ADC) Block

Throughput vs. Resolution Trade-Off in ADC

•Throughput: ADC’s speed, e.g., 1000 Hz sampling for a 250 Hz

signal, or 3 MHz in oversampling.

•Resolution: Bit depth (e.g., 1-bit or 13-bit), where more bits mean

better accuracy but higher complexity.

•Trade-Off: High speed with low bits (e.g., 1-bit) spreads noise,

improves SNR; high bits at high speed are costly.

Questions

1.What are the three main steps in

analog-to-digital conversion

(ADC)?

2.What does the quantizer do in an

ADC system?

3.What is the trade-off between

throughput and resolution in

ADCs?

Quantization

•Quantization is the process of mapping a continuous range of

values (amplitudes of an analog signal) to a finite set of discrete

levels.

•The main purpose of quantization is to enable the representation

of analog signals in a form that can be easily processed, stored,

and transmitted in digital systems.

Quantization Noise(Error)

•Occurs during analog-to-digital conversion (ADC) of signals.

•Caused by rounding analog values to the nearest digital level.

•Appears as small, random errors (noise) in the digitized signal.

•More noticeable with low-resolution ADCs (fewer bits).

•Can distort low-amplitude signals like EEG or ECG.

•Not easily removed by filtering; reduces signal accuracy.

Signal to Quantization Noise Ratio

•The signal-to-quantization-noise ratio (SQNR) measures how much the

signal stands above this noise, improving by about 6 dB for each extra

bit of resolution in the ADC.

•For example, moving from a 12-bit to a 16-bit ADC in an ECG application

adds 4 bits, boosting SQNR by 24 dB, ensuring finer detail in heart

signals.

•This enhancement makes a 16-bit ADC far superior for capturing subtle

ECG changes compared to a 12-bit one, critical for accurate medical

diagnosis.

Fixed- vs floating-point ADC output

•Digital signal processing can be separated into two categories - fixed

point and floating point.

•These designations refer to the format used to store and manipulate

numeric representations of data.

•Fixed-point DSPs are designed to represent and manipulate integers

–positive and negative whole numbers –via a minimum of 16 bits,

yielding up to 65,536 possible bit patterns (216).

•A fixed-point ADC output keeps numbers simple with a set range, like

a 12-bit Successive Approximation Register(SAR) used in an ECG

patch to track heartbeats affordably.

Fixed- vs floating-point ADC output

•Floating-point DSPs represent and manipulate rational numbers via a

minimum of 32 bits in a manner similar to scientific notation.

•The number is represented with a mantissa and an exponent (e.g., A x

2B, where 'A' is the mantissa and ‘B’ is the exponent), yielding up to

4,294,967,296 possible bit patterns (232).

•A floating-point ADC output, such as a 24-bit sigma-delta for EEG,

flexes to handle big and small brain signals with great detail.

Fixed- vs floating-point ADC output

•The term ‘fixed point’ refers to the corresponding manner in which

numbers are represented, with a fixed number of digits after, and

sometimes before, the decimal point.

•With floating-point representation, the placement of the decimal

point can ‘float’ relative to the significant digits of the number.

•The 24-bit option captures tiny EEG changes, while the 12-bit option

works fine for the steadier ECG signals in a patch.

•It’s all about matching the ADC to what the biosignal needs without

overcomplicating things.

Anti-Aliasing (pre-sample) filters

•Anti-aliasing is a technique used in digital signal processing and

imaging to prevent or reduce aliasing, a distortion that occurs when

a continuous signal is sampled at a rate too low to accurately

represent its frequency content

•Prevents aliasing by removing high frequencies before sampling.

•A low-pass filter blocks frequencies above the Nyquist rate (fs/2).

•Sharp cutoff to ensure only capturable frequencies remain.

•This ensures the sampled signal stays clear by softening frequencies

that could cause distortion.

Filter Selection for Biosignals

•Filter selection for biosignals depends on their frequency range to

capture the right details without noise.

•For ECG and PPG signals, which range from 0.5 to 40 Hz, a low-pass

filter at 100 Hz keeps the heart’s rhythm clear while cutting out high-

frequency interference.

•EMG signals, with a wider band up to 500 Hz, need a low-pass filter

at 1 kHz to include muscle activity without distortion.

•This tailored approach ensures each biosignal is accurately recorded

for medical use.

Questions

1.What causes quantization noise in

ADCs?

2.How does increasing ADC

resolution affect the signal-to-

quantization-noise ratio (SQNR)?

3.Why are anti-aliasing filters applied

before sampling biosignals?

Summary

•Continuous-time signals (e.g., ECG) are defined at all time points,

while discrete-time signals are sampled at intervals.

•To avoid aliasing, signals must be sampled at least twice their

highest frequency, as per the Nyquist Theorem.

•ADC converts analog signals to digital through sampling,

quantization, and encoding.

•Quantization introduces noise, but increasing ADC resolution

improves accuracy (higher SQNR).

Summary

•Fixed-point ADCs are simple and used for signals like ECG,

while floating-point ADCs offer greater precision for complex

signals like EEG.

•Anti-aliasing filters remove high-frequency noise before

sampling, with filter choice depending on the signal type (e.g.,

ECG or EMG).

Recommended Texts

•Rangayyan, R. M. (2015). Biomedical signal analysis: A case-study

approach (2nd ed.). IEEE Press Series in Biomedical Engineering.

Wiley–IEEE Press. ISBN: 978-0-470-01139-6.

•Palaniappan, R. (2011). Biological signal analysis. University of Essex

•Delgutte B (2007). Course materials for HST.582J / 6.555J / 16.456J,

Biomedical Signal and Image Processing, Spring 2007. MIT

OpenCourseWare, Massachusetts Institute of Technology.