2.1 Sampling and quantization

Sampling and quantization are fundamental processes in digital signal processing. They bridge the gap between analog and digital worlds, allowing us to represent continuous signals as discrete values. These techniques are crucial for processing, storing, and transmitting information in modern digital systems.

Understanding sampling and quantization is essential for designing effective digital systems. From the Nyquist sampling theorem to oversampling techniques, these concepts form the foundation for converting analog signals to digital form and back again. They impact everything from audio processing to telecommunications and beyond.

Sampling of continuous-time signals

• Sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing values at regular intervals
• Sampling allows for the digital processing, storage, and transmission of analog signals, which is essential in modern signal processing applications

Nyquist sampling theorem

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• States that a continuous-time signal can be perfectly reconstructed from its samples if the sampling rate is at least twice the highest frequency component of the signal
• The minimum sampling rate required to avoid aliasing is called the Nyquist rate, given by $f_s \geq 2f_{max}$, where $f_s$ is the sampling rate and $f_{max}$ is the highest frequency component of the signal
• If the sampling rate is lower than the Nyquist rate, aliasing occurs, leading to distortion and loss of information

Sampling rate vs signal bandwidth

• The bandwidth of a signal refers to the range of frequencies present in the signal
• To accurately represent a signal, the sampling rate must be chosen based on the signal's bandwidth
• According to the Nyquist sampling theorem, the sampling rate should be at least twice the signal's bandwidth to avoid aliasing

Aliasing in undersampled signals

• Aliasing occurs when a signal is sampled at a rate lower than the Nyquist rate
• In the frequency domain, aliasing manifests as high-frequency components folding back into the lower-frequency range, causing distortion and ambiguity
• Aliased frequency components cannot be distinguished from the original signal components, leading to irreversible loss of information

Anti-aliasing filters for sampling

• Anti-aliasing filters are low-pass filters used to limit the bandwidth of a signal before sampling
• By attenuating frequency components above half the sampling rate, anti-aliasing filters prevent aliasing and ensure accurate signal representation
• Ideal anti-aliasing filters have a sharp cutoff at the Nyquist frequency, but practical filters exhibit a transition band and passband ripple

Sampling of discrete-time signals

• Discrete-time signals are sequences of values defined at integer time indices
• Sampling of discrete-time signals involves changing the sampling rate, which can be achieved through upsampling or downsampling

Upsampling vs downsampling

• Upsampling increases the sampling rate of a discrete-time signal by inserting zeros between the original samples
• Downsampling reduces the sampling rate of a discrete-time signal by keeping only every $M$-th sample, where $M$ is the downsampling factor
• Upsampling expands the signal in the time domain and compresses the frequency spectrum, while downsampling compresses the signal in the time domain and expands the frequency spectrum

Interpolation in upsampling

• After upsampling, the inserted zeros create spectral replicas in the frequency domain
• Interpolation is the process of filling in the missing samples by applying a low-pass filter to remove the spectral replicas and reconstruct the signal
• Common interpolation methods include zero-order hold, linear interpolation, and sinc interpolation

Decimation in downsampling

• Decimation is the process of reducing the sampling rate by first applying an anti-aliasing filter to the signal and then discarding samples
• The anti-aliasing filter is necessary to prevent aliasing when downsampling, as the reduced sampling rate may not satisfy the Nyquist criterion
• Decimation is often used to reduce the computational complexity and data rate of signal processing systems

Resampling of discrete-time signals

• Resampling is the process of changing the sampling rate of a discrete-time signal by a rational factor $L/M$
• Resampling can be achieved by first upsampling the signal by a factor of $L$, then applying an anti-aliasing filter, and finally downsampling the filtered signal by a factor of $M$
• Resampling is useful for sample rate conversion between different systems or for efficient signal processing at lower sampling rates

Quantization of sampled signals

• Quantization is the process of mapping a continuous range of values to a discrete set of values
• In digital signal processing, quantization is necessary to represent the amplitude of sampled signals using a finite number of bits

Uniform vs non-uniform quantization

• Uniform quantization divides the input range into equally spaced intervals, with each interval assigned a unique discrete value
• Non-uniform quantization uses unequally spaced intervals, with smaller intervals assigned to more frequently occurring or more important signal values
• Non-uniform quantization can be achieved using companding techniques, such as $\mu$-law or A-law, which compress the signal before uniform quantization and expand it after quantization

Quantization noise

• Quantization noise is the error introduced when a continuous-valued signal is approximated by a discrete-valued signal
• Quantization noise is caused by the rounding or truncation of the signal values to the nearest quantization levels
• The magnitude of quantization noise depends on the number of quantization levels and the signal's amplitude distribution

Signal-to-quantization-noise ratio (SQNR)

• SQNR is a measure of the quality of a quantized signal, expressed as the ratio of the signal power to the quantization noise power
• For a uniform quantizer with $N$ bits, the SQNR is given by $SQNR = 6.02N + 1.76$ dB, assuming a full-scale sinusoidal input signal
• Increasing the number of quantization bits improves the SQNR, but also increases the data rate and computational complexity

Dithering for quantization noise reduction

• Dithering is the process of adding a small amount of random noise to a signal before quantization
• Dithering helps to randomize the quantization error, reducing the perception of quantization noise and avoiding harmonic distortion
• Common dithering techniques include rectangular dither, triangular dither, and noise shaping, which shapes the spectrum of the quantization noise to minimize its audibility

Pulse code modulation (PCM)

• PCM is a digital representation of an analog signal, where the signal is sampled at regular intervals and each sample is quantized to a discrete value
• PCM is widely used in digital audio and telecommunications systems

PCM encoding vs decoding

• PCM encoding involves sampling an analog signal, quantizing the samples, and converting the quantized values into a digital bitstream
• PCM decoding reconstructs the analog signal from the digital bitstream by converting the bits back to quantized values and then applying a low-pass filter to smooth the signal
• The encoding and decoding processes are designed to minimize the loss of information and maintain the signal's quality

PCM bit rate vs quantization levels

• The PCM bit rate is the number of bits transmitted per second, given by the product of the sampling rate and the number of bits per sample
• Increasing the number of quantization levels (i.e., the number of bits per sample) improves the signal-to-quantization-noise ratio (SQNR) but also increases the bit rate
• Common PCM formats include 8-bit, 16-bit, 24-bit, and 32-bit, with higher bit depths providing better audio quality but requiring more storage and transmission bandwidth

Companding in PCM

• Companding is a technique used to improve the SQNR of PCM systems by applying non-uniform quantization
• The two most common companding algorithms are $\mu$-law (used in North America and Japan) and A-law (used in Europe and the rest of the world)
• Companding involves compressing the signal before quantization and expanding it after quantization, which allocates more quantization levels to lower-amplitude signals and fewer levels to higher-amplitude signals

Oversampling techniques

• Oversampling is the process of sampling a signal at a rate much higher than the Nyquist rate
• Oversampling techniques are used to improve the resolution and signal-to-noise ratio (SNR) of analog-to-digital converters (ADCs) and digital-to-analog converters (DACs)

Oversampling ADC vs Nyquist-rate ADC

• An oversampling ADC samples the input signal at a rate much higher than the Nyquist rate, typically by a factor of 2 to 256
• Oversampling ADCs have a simpler anti-aliasing filter requirement compared to Nyquist-rate ADCs, as the oversampling pushes the aliasing components further away from the signal band
• Oversampling ADCs also benefit from increased SNR due to the spreading of quantization noise over a wider frequency range

Sigma-delta modulation

• Sigma-delta modulation is a widely used oversampling technique in ADCs and DACs
• It consists of an integrator, a comparator (1-bit quantizer), and a feedback loop with a 1-bit DAC
• The integrator accumulates the difference between the input signal and the feedback signal, while the comparator produces a 1-bit output based on the sign of the integrator output
• The 1-bit DAC in the feedback loop helps to shape the quantization noise, pushing it to higher frequencies

Noise shaping in oversampling

• Noise shaping is a technique used in oversampling ADCs and DACs to redistribute the quantization noise across the frequency spectrum
• By using a high-order loop filter in the sigma-delta modulator, noise shaping pushes the quantization noise to higher frequencies, where it can be easily filtered out
• Noise shaping improves the SNR in the signal band at the expense of increased noise at higher frequencies, which are eventually removed by a digital low-pass filter

Practical considerations

• When implementing sampling and quantization in real-world systems, several practical factors must be considered to ensure optimal performance and efficiency

Finite word length effects

• Finite word length effects arise due to the limited precision of digital systems, which use a fixed number of bits to represent signals and coefficients
• Quantization of coefficients in digital filters and other signal processing algorithms can lead to deviations from the ideal response, such as increased passband ripple, reduced stopband attenuation, and shifts in pole/zero locations
• Roundoff noise, caused by rounding or truncating arithmetic operations, can accumulate and degrade the signal quality, especially in recursive systems like IIR filters

Computational complexity of sampling and quantization

• The computational complexity of sampling and quantization algorithms directly impacts the power consumption, processing time, and hardware requirements of the system
• Oversampling techniques, such as sigma-delta modulation, require high-speed digital signal processing and can be computationally intensive
• Efficient implementation of interpolation and decimation filters, as well as quantization and dithering algorithms, is crucial for real-time applications and power-constrained devices

Hardware implementation of sampling and quantization

• Hardware implementation of sampling and quantization involves the design of analog-to-digital converters (ADCs), digital-to-analog converters (DACs), and associated circuitry
• The choice of ADC and DAC architectures (e.g., flash, successive approximation, pipelined, sigma-delta) depends on the application requirements, such as sampling rate, resolution, power consumption, and cost
• Anti-aliasing filters and reconstruction filters must be carefully designed to minimize distortion and ensure proper band-limiting of the signal
• Clock jitter, thermal noise, and other circuit-level impairments can degrade the performance of sampling and quantization systems, requiring careful design and layout techniques to mitigate their effects