Discrete and Continuous Signal Processing#
Signal processing can be broadly categorized into two types based on the nature of the signals being processed: continuous and discrete.
Continuous Signal Processing#
Continuous signal processing deals with signals that are defined for every point in time. These signals, often referred to as analog signals, are represented mathematically as functions of a continuous variable (typically time). Examples include audio signals captured by a microphone, electrical signals in circuits, and temperature readings over time.
Mathematically, a continuous signal is represented as \(x(t)\), where \(t\) is a continuous variable representing time. These signals are processed using analog techniques, which involve operations such as filtering, modulation, and amplification.
Discrete Signal Processing#
Discrete signal processing, on the other hand, deals with signals that are defined only at discrete points in time. These signals, known as digital signals, are obtained by sampling continuous signals at regular intervals. Discrete signal processing techniques are implemented using digital systems, such as computers and digital signal processors (DSPs).
A discrete sequence \(\{ x_n \}\) is a sequence of numbers \(\cdots, x_{-2}, x_{-1}, x_{0}, x_{1}, x_{2}, \cdots \), where \(x_n\) denotes the \(n\)-th number in the sequence with \(n \in \mathcal{Z}\). This sequence maps integer numbers onto real (or complex) numbers. For simplicity, this notation is often abbreviated as \(\{ x_n \}\).
→ Some authors write \(x[n]\) and others \( x(n) \).
A discrete sequence \(\{ x_n \}\) samples a continuous function \(x(t)\) as
where \(t_s\) is the sampling period and \(f_s = \frac{1}{t_s}\) is the sampling frequency.
Conclusion#
Both continuous and discrete signal processing have their unique applications and advantages. Continuous signal processing is crucial for handling real-world analog signals, while discrete signal processing is essential for digital systems and modern computational techniques. Understanding the distinction and relationship between these two types of signal processing is fundamental for effectively analyzing and manipulating signals in various domains.