Complex Sinusoids Archives | Page 3 of 4

March 21, 2022 DSP

A Visualization of Causality and Stability in z-Transform

Most of the books and resources explain the z-Transform as a mathematical concept rather than a signal processing idea. Today I will provide a simple explanation of how the z-Transform helps in determining whether a system is causal and stable. I hope that this visual approach will help my readers learn this concept in a better manner. The z-Transform For a discrete-time signal $h[n]$ (that is the impulse response of a system), the z-Transform is defined as \begin{equation}\label{equation-z-transform} H(z) = \sum _{n=-\infty}^{\infty} h[n]z^{-n} \end{equation} Then, $z$ is a complex number and hence can be written as \[ z = re^{j\omega}

March 25, 2024 DSP / Miscellaneous

Why the Constant e Arises in Complex Plane as a Rotation

In the tutorial on how complex numbers arose, we asked three questions. The first two were answered in the same article while the answer to the third question, repeated below, is explained here. Why is the expression $e^{i \theta}$ a rotation of 1 by $\theta$ radians on a unit circle? Is it possible to make sense out of a number like $2.71828^{\sqrt{-1}\cdot\theta}$? The constant e is a special number discovered by Jacob Bernoulli while studying compound interests. It appears in many other forms as well which are all related to each other but that topic is a complete account in

Signal frequency and analysis frequency are not the same, resulting in DFT leakage and spectrum sampling away from the peak. The spectrum shown here is oversampled by 3 through 3x zero-padded FFT

October 16, 2026 DSP

Peak Interpolation for Sub-Bin Accuracy in Radar and Communications

In signal processing applications, one frequently encounters the problem of estimating the frequency of a complex sinusoid embedded in noise. For example, a DSP engineer will encounter this problem in the delayed echo of an FMCW radar that directly translates into target range, or in carrier frequency estimation of a wireless communication system after timing synchronization has already been achieved (see Chapter 6 for details of how modulation removal exposes the underlying complex sinusoid). Beyond communications and radar, this frequency estimation is also heavily relied upon in digital pitch tracking, speech synthesis, active and passive sonar detection, Doppler ultrasound and

Intuition behind multiplying two complex numbers

March 14, 2025 Wireless and SDR

Intuitive Reason behind Multiplication and Division of Complex Numbers

Any introduction to complex numbers and their operations follows a common pattern: the formulas are given without building any intuition. If you have ever wondered about why $j=\sqrt{-1}$, you can read about the origin of complex numbers. In this article, I will explain the intuitive reason behind why a product of two complex numbers multiplies their magnitudes and adds their angles, and a division of two complex numbers divides their magnitudes and subtracts their angles. Let us start with the multiplication. The division scenario can be analogously derived with inverse operations. Multiplication of Complex Numbers The intuition behind multiplying two

Symbolic representation of linear phase rotation that changes with frequency index

May 14, 2019 DSP

Effect of Time Shift in Frequency Domain

Children usually ask questions like “How many hours have passed?” And they have no idea about the start time to be taken as a reference. Just like the zero of a measuring tape, a zero reference for time plays a crucial role in analyzing the signal behaviour in time and frequency domains. Until now, we assumed that reference time $0$ coincides with the start of a sine and a cosine wave to understand the frequency domain. Later, we will deal with symbol timing synchronization problem in single-carrier systems and carrier frequency synchronization problem in multicarrier systems, both of which address