🔌Intro to Electrical Engineering Unit 20 – Sampling and Discrete-Time Signals

Key Concepts and Definitions

• Sampling involves converting a continuous-time signal into a discrete-time signal by taking samples at regular intervals
• Discrete-time signals are represented by a sequence of numbers, where each number corresponds to a specific time instant
  • Denoted as x[n], where n is an integer representing the sample index
• Sampling period (T) is the time interval between consecutive samples in a discrete-time signal
• Sampling frequency (fs) is the number of samples taken per second, measured in Hertz (Hz)
  • Related to the sampling period by fs = 1/T
• Nyquist rate is the minimum sampling frequency required to avoid aliasing and accurately reconstruct the original signal
• Aliasing occurs when a signal is sampled at a rate lower than the Nyquist rate, resulting in frequency components that are indistinguishable from lower-frequency components
• Quantization is the process of representing a continuous range of values with a finite set of discrete values
  • Introduces quantization noise, which affects the signal-to-noise ratio (SNR)

Sampling Theory Fundamentals

• Sampling theorem, also known as the Nyquist-Shannon sampling theorem, states that a band-limited signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• Band-limited signals have a finite bandwidth, meaning their frequency components are limited to a specific range
• Ideal sampling is a mathematical abstraction that involves multiplying a continuous-time signal by a train of impulses spaced at the sampling period
  • Results in a sequence of impulses with amplitudes equal to the signal values at the sampling instants
• Practical sampling uses sample-and-hold circuits to capture and maintain the signal value for a short duration before the next sample is taken
• Reconstruction of a sampled signal involves interpolating between the discrete samples to obtain a continuous-time approximation of the original signal
  • Ideal reconstruction uses a sinc interpolation function, which is the inverse Fourier transform of an ideal low-pass filter
• Sampling and reconstruction processes introduce errors due to non-ideal characteristics of real-world systems, such as non-ideal filters and quantization noise

Discrete-Time Signal Representation

• Discrete-time signals can be represented mathematically using sequences, where each element corresponds to a specific time index
  • x[n] represents the value of the signal at time index n
• Unit sample sequence, also known as the unit impulse or Kronecker delta function, is defined as:
  • δ[n] = 1 for n = 0, and δ[n] = 0 for n ≠ 0
• Unit step sequence, denoted as u[n], is defined as:
  • u[n] = 1 for n ≥ 0, and u[n] = 0 for n < 0
• Discrete-time exponential sequence is defined as:
  • x[n] = a^n, where a is a constant
• Sinusoidal sequences are represented by discrete-time sinusoids:
  • x[n] = A cos(ω0n + φ), where A is the amplitude, ω0 is the discrete-time frequency, and φ is the phase
• Discrete-time Fourier transform (DTFT) is used to analyze the frequency content of discrete-time signals
  • Converts a discrete-time signal into a continuous function of frequency
• Z-transform is a generalization of the DTFT, providing a tool for analyzing and manipulating discrete-time signals and systems
  • Converts a discrete-time signal into a complex function of a complex variable z

Sampling Rate and Nyquist Theorem

• Nyquist rate, also known as the Nyquist frequency, is the minimum sampling frequency required to avoid aliasing when sampling a band-limited signal
  • Equal to twice the highest frequency component in the signal
• Undersampling occurs when the sampling frequency is lower than the Nyquist rate
  • Results in aliasing, where high-frequency components appear as lower-frequency components in the sampled signal
• Oversampling involves sampling a signal at a rate higher than the Nyquist rate
  • Provides better noise immunity and allows for simpler anti-aliasing filter designs
• Nyquist theorem states that a band-limited signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component in the signal
• Folding frequency, also known as the Nyquist frequency, is half the sampling frequency
  • Represents the highest frequency that can be unambiguously represented in a sampled signal
• Sampling rate conversion techniques, such as upsampling and downsampling, are used to change the sampling rate of a discrete-time signal
  • Upsampling increases the sampling rate by inserting zeros between samples
  • Downsampling reduces the sampling rate by discarding samples

Aliasing and Anti-Aliasing Techniques

• Aliasing occurs when a signal is sampled at a rate lower than the Nyquist rate, causing high-frequency components to be misinterpreted as lower-frequency components
  • Results in distortion and loss of information in the reconstructed signal
• Aliasing can be prevented by ensuring that the signal is band-limited and sampled at a rate higher than the Nyquist rate
• Anti-aliasing filters are used to limit the bandwidth of a signal before sampling
  • Low-pass filters with a cutoff frequency below the Nyquist frequency are commonly used
• Analog anti-aliasing filters are placed before the analog-to-digital converter (ADC) to remove high-frequency components that would cause aliasing
• Digital anti-aliasing filters, such as decimation filters, are used to reduce the sampling rate while preventing aliasing
  • Involve low-pass filtering followed by downsampling
• Oversampling and noise shaping techniques can be used to reduce the complexity of anti-aliasing filters
  • Oversampling spreads the quantization noise over a wider frequency range
  • Noise shaping pushes the quantization noise to higher frequencies, which can be easily filtered out

Quantization and Signal-to-Noise Ratio

• Quantization is the process of representing a continuous range of values with a finite set of discrete values
  • Introduces quantization error, which is the difference between the original value and its quantized representation
• Quantization step size, denoted as Δ, is the difference between two consecutive quantization levels
  • Smaller step sizes result in higher resolution and lower quantization noise
• Quantization noise is the error introduced by the quantization process
  • Modeled as a random variable uniformly distributed between -Δ/2 and Δ/2
• Signal-to-quantization-noise ratio (SQNR) is a measure of the ratio between the signal power and the quantization noise power
  • Expressed in decibels (dB) as: SQNR = 10log10(Psignal / Pnoise)
• Effective number of bits (ENOB) is a measure of the actual resolution of an ADC, considering the effects of quantization noise and other non-idealities
  • Calculated as: ENOB = (SQNR - 1.76) / 6.02
• Dither is a technique used to randomize quantization error and improve the perceived signal quality
  • Involves adding a small amount of random noise to the signal before quantization
• Oversampling and noise shaping techniques can be used to improve the SQNR and ENOB of a quantized signal
  • Oversampling reduces the quantization noise power by spreading it over a wider frequency range
  • Noise shaping redistributes the quantization noise to higher frequencies, where it is less perceptible

Applications in Digital Signal Processing

• Digital signal processing (DSP) relies heavily on sampling and discrete-time signal representation
• Audio and speech processing applications, such as digital audio workstations and voice recognition systems, use sampling to convert analog audio signals into discrete-time representations
  • Typical sampling rates for audio include 44.1 kHz (CD quality), 48 kHz (professional audio), and 96 kHz (high-resolution audio)
• Image and video processing applications, such as digital cameras and video codecs, use sampling to convert continuous-time visual information into discrete-time formats
  • Image sensors in digital cameras sample the continuous-time light intensity at discrete spatial locations
  • Video frames are sampled at a specific frame rate, such as 30 frames per second (fps) or 60 fps
• Wireless communication systems, such as cellular networks and Wi-Fi, use sampling to convert analog signals into discrete-time representations for digital transmission
  • Baseband signals are sampled at a rate determined by the signal bandwidth and the desired signal quality
• Radar and sonar systems use sampling to convert continuous-time reflected signals into discrete-time representations for processing and analysis
  • Sampling rates are determined by the desired range resolution and the maximum range of the system
• Biomedical signal processing, such as in electrocardiography (ECG) and electroencephalography (EEG), uses sampling to convert physiological signals into discrete-time representations for analysis and diagnosis
  • Sampling rates are chosen based on the frequency content of the physiological signals and the desired temporal resolution

Practical Examples and Problem Solving

• Example 1: Determine the Nyquist rate and the minimum sampling frequency for an analog signal with a bandwidth of 5 kHz.
  • Solution: The Nyquist rate is twice the highest frequency component, so the Nyquist rate is 2 × 5 kHz = 10 kHz. The minimum sampling frequency is equal to the Nyquist rate, so the minimum sampling frequency is 10 kHz.
• Example 2: An analog signal is sampled at 8 kHz. What is the folding frequency and the maximum frequency that can be unambiguously represented in the sampled signal?
  • Solution: The folding frequency is half the sampling frequency, so the folding frequency is 8 kHz / 2 = 4 kHz. The maximum frequency that can be unambiguously represented is equal to the folding frequency, so it is also 4 kHz.
• Example 3: Calculate the signal-to-quantization-noise ratio (SQNR) and the effective number of bits (ENOB) for a 12-bit ADC with a full-scale voltage range of 3.3V, assuming a sinusoidal input signal with an amplitude of 1V.
  • Solution: The quantization step size is Δ = 3.3V / 2^12 ≈ 0.806mV. The signal power is Psignal = (1V)^2 / 2 = 0.5W. The quantization noise power is Pnoise = Δ^2 / 12 ≈ 54.02pW. The SQNR is SQNR = 10log10(Psignal / Pnoise) ≈ 69.66dB. The ENOB is ENOB = (SQNR - 1.76) / 6.02 ≈ 11.28 bits.
• Example 4: A continuous-time signal with a bandwidth of 2 kHz is sampled at 5 kHz. Determine if aliasing occurs and, if so, identify the aliased frequency components.
  • Solution: The Nyquist rate is 2 × 2 kHz = 4 kHz. Since the sampling frequency (5 kHz) is greater than the Nyquist rate, aliasing does not occur in this case.
• Example 5: Design an anti-aliasing filter for a data acquisition system with a sampling rate of 10 kHz. The filter should have a passband of 0-4 kHz with a maximum passband ripple of 0.1 dB and a stopband starting at 4.5 kHz with a minimum attenuation of 60 dB.
  • Solution: To meet the given specifications, a low-pass filter with a sharp transition band is required. An elliptic (Cauer) filter or a high-order Chebyshev Type II filter could be suitable choices. The filter order and cutoff frequency can be determined using filter design tools or tables based on the desired passband ripple and stopband attenuation. The designed filter should be implemented as an analog anti-aliasing filter before the ADC to prevent aliasing.