12.1 Introduction to Discrete-Time Systems

Discrete-time systems are the backbone of digital signal processing. They convert continuous signals into sequences of numbers, allowing for efficient analysis and manipulation using computers and digital devices.

Sampling is key in this process, turning continuous signals into discrete ones at regular intervals. Understanding sampling frequency, aliasing, and the Nyquist theorem is crucial for accurately representing and processing signals in the digital domain.

Sampling in Discrete-Time Systems

The Fundamentals of Sampling

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• Sampling converts a continuous-time signal into a discrete-time signal by capturing the signal's amplitude at regular intervals called the sampling period (T) or sampling interval
• The sampling frequency (fs) or sampling rate represents the number of samples taken per second, measured in Hertz (Hz) or samples per second
  • fs is the reciprocal of the sampling period: fs = 1/T
• The sampled signal is a sequence of discrete values, denoted as x[n], where n is an integer representing the sample index or discrete-time instant
  • For example, if a continuous-time signal x(t) is sampled every T seconds, the resulting discrete-time signal would be x[n] = x(nT), where n = 0, 1, 2, ...

The Importance of Sampling in Discrete-Time Systems

• Sampling is essential in discrete-time systems because it allows the representation and processing of continuous-time signals using digital devices (computers, digital signal processors)
• Sampling enables the application of discrete-time signal processing techniques, such as:
  • Digital filtering
  • Spectral analysis using the Discrete Fourier Transform (DFT)
  • Implementation of algorithms for signal compression, denoising, and feature extraction
• Sampled signals can be stored, transmitted, and manipulated more efficiently than continuous-time signals due to their discrete nature and finite representation

Continuous vs Discrete-Time Signals

Defining Characteristics

• Continuous-time signals are defined for all values of time and have an infinite number of values within any time interval
• Discrete-time signals are defined only at specific time instants and have a finite number of values
• The independent variable for continuous-time signals is time (t), which is a real number
• The independent variable for discrete-time signals is the sample index (n), which is an integer

Representation and Notation

• Continuous-time signals are typically represented by functions, such as x(t)
  • Example: A sinusoidal signal x(t) = sin(2πft), where f is the frequency in Hz
• Discrete-time signals are represented by sequences, such as x[n]
  • Example: A discrete-time exponential signal x[n] = a^n, where a is a constant and n is the sample index
• Continuous-time signals are often depicted as waveforms on a time-domain plot
• Discrete-time signals are usually represented as stem plots or connected dots on a sample index-domain plot

Mathematical Operations and Analysis

• Mathematical operations and analysis techniques differ between continuous-time and discrete-time signals
• Continuous-time signals use concepts like differential equations, Laplace transforms, and Fourier transforms
• Discrete-time signals use concepts like difference equations, z-transforms, and discrete-time Fourier transforms
  • Example: The z-transform of a discrete-time signal x[n] is defined as X(z) = Σ x[n]z^(-n), where z is a complex variable
• Discrete-time signals can be processed using efficient algorithms implemented on digital hardware or software platforms

Effects of Sampling on Signals

Information Loss and Signal Reconstruction

• Sampling a continuous-time signal results in a loss of information between the sampled points, as the signal's values are known only at the sampling instants
• The choice of sampling frequency affects the accuracy of the sampled signal's representation
  • Higher sampling frequencies provide a more accurate representation of the original continuous-time signal
• The reconstruction of a continuous-time signal from its sampled version involves interpolation techniques, such as:
  • Zero-order hold (ZOH): Holding each sample value constant until the next sample
  • Linear interpolation: Connecting adjacent samples with straight lines
  • Higher-order interpolation methods (spline interpolation): Estimating the signal's values between the sampled points using smooth curves

Aliasing and Undersampling

• Undersampling occurs when the sampling frequency is too low to capture the high-frequency components of the signal adequately
• Aliasing is a consequence of undersampling, where high-frequency components of the signal are misrepresented as lower-frequency components in the sampled signal
  • Example: If a sinusoidal signal with a frequency of 100 Hz is sampled at 150 Hz (below the Nyquist rate), it will appear as a 50 Hz sinusoid in the sampled signal due to aliasing
• Aliasing can cause distortion and loss of information in the reconstructed signal
• To prevent aliasing, the sampling frequency must be chosen according to the Nyquist-Shannon sampling theorem (discussed in the next section)

Sampling Theorem for Aliasing Prevention

The Nyquist-Shannon Sampling Theorem

• The Nyquist-Shannon sampling theorem states that a continuous-time signal can be perfectly reconstructed from its sampled version if the sampling frequency is at least twice the highest frequency component present in the signal
• The minimum sampling frequency required to avoid aliasing is called the Nyquist rate, which is equal to twice the signal's bandwidth (fB)
  • Nyquist rate: fs ≥ 2fB
  • Example: If a signal has a bandwidth of 500 Hz, the minimum sampling frequency to avoid aliasing would be 1000 Hz (twice the bandwidth)
• If the sampling frequency is less than the Nyquist rate, aliasing occurs, and the original signal cannot be accurately reconstructed from the sampled signal

Anti-Aliasing Filters and Band-Limiting

• To ensure accurate signal representation and avoid aliasing, the continuous-time signal should be band-limited before sampling
• An anti-aliasing filter, typically a low-pass filter, is used to remove frequency components above half the sampling frequency
  • The anti-aliasing filter should have a cutoff frequency less than or equal to half the sampling frequency (fc ≤ fs/2)
• Band-limiting the signal ensures that the Nyquist-Shannon sampling theorem is satisfied, and the signal can be reconstructed without aliasing

Techniques for Dealing with Nyquist Rate Limitations

• When the Nyquist rate is not achievable due to hardware limitations, techniques such as oversampling and noise shaping can be employed
• Oversampling involves sampling the signal at a higher frequency than the Nyquist rate and then downsampling the signal after applying digital filtering
  • Oversampling helps to distribute quantization noise over a wider frequency range, improving the signal-to-noise ratio
• Noise shaping is a technique used in oversampling analog-to-digital converters (ADCs) to push the quantization noise to higher frequencies, which can be removed by digital filtering
  • Example: Delta-sigma modulation is a noise shaping technique commonly used in audio ADCs to achieve high resolution and low noise