Convolution, Fourier Transform, S-Transform, and Z-Transform

In this article, we will discuss fundamental concepts related to signal forms, convolution, Fourier transform, s-transform, and z-transform. First, let's explore the basic forms of signals. Signals can be continuous or discrete. Continuous signals vary continuously over time, while discrete signals change at distinct time intervals. Next, we introduce convolution—a mathematical operation that combines two functions to produce a third function. In practical implementations, convolution can be computed using algorithms like overlap-add or directly via functions such as numpy.convolve() in Python. Following this, we examine the Fourier transform, which converts signals from the time domain to the frequency domain. Key functions like FFT (Fast Fourier Transform) libraries (e.g., fft in MATLAB or scipy.fft in Python) enable efficient computation for both analysis and filtering applications. We then delve into the s-transform and z-transform. The s-transform operates in the complex plane for continuous-time signals, often utilized in control system design with Laplace domain analysis. The z-transform applies to discrete-time signals in the complex plane, essential for digital signal processing (DSP) and implemented using tools like MATLAB’s ztrans function or symbolic computation in Python. Understanding these concepts provides a solid foundation for signal processing, enabling effective algorithm development and system modeling.