Complex Sinusoids Archives | Page 3 of 4

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

December 22, 2017 DSP

Interpreting Time Domain Derivative in Frequency Domain

Although this article explains the concepts in terms of mathematical constants e and j as well as integration, my book on SDR steers clear of the complex notation and integrals to describe the underlying concepts from the ground up to an advanced level. One of the properties of Fourier Transform is that the derivative of a signal in time domain gets translated to multiplication of the signal spectrum by $j2\pi f$ in frequency domain. This property is usually derived as follows. For a signal $s(t)$ with Fourier Transform $S(f)$ \begin{equation*} s(t) = \frac{1}{2\pi}\int \limits _{-\infty}^{+\infty} S(f) e^{j2\pi ft}df, \end{equation*} we

November 30, 2023 DSP / Localization

FMCW Radar Part 3 – Design Guidelines

The Bloom’s Taxonomy describes the levels of mastery one attains in a field. Its last two stages are Synthesis and Evaluation. This is where the masters can be differentiated from the experts. In a job interview, for example, a good technique to judge a candidate’s ability is to ask them where the system in question breaks. A little learning is a dangerous thing Drink deep, or taste not the Pierian spring There shallow draughts intoxicate the brain And drinking largely sobers us again While the first two parts of the FMCW radar series addressed the lower levels, Part 3 is

Beat frequency sinusoid and its spectrum

October 31, 2023 DSP

Spectrum of a Sinusoid

In this article, we derive the spectrum of a complex sinusoid that acts as the basis for all spectra. In fact, the very definition of the Fourier Transform, whether continuous or discrete, comes from the perspective of a complex sinusoid. Therefore, exploring this derivation will be useful in everything else we learn about DSP. For an in-depth understanding of complex signals and I/Q processing, you can read the following two articles (the option of downloading them as PDF is available). The origin of complex numbers and signals I/Q signal processing Let us start with the continuous-time case. A Continuous-Time Sinusoid

August 15, 2022 DSP

Goertzel Algorithm – Evaluating DFT without DFT

The Discrete Fourier Transform (DFT) computes the contribution of $N$ sinusoids that come together to form any input signal. However, in some applications, we are only interested in contributions from one or a few sinusoids This is where the Goertzel algorithm, proposed by Gerald Goertzel in 1958, comes in. The Goertzel algorithm evaluates the individual terms of the DFT in an efficient manner. We explain its derivation and implementation with the help of DTMF signals. DTMF Signal Generation In the early days of telephone, you could not call anyone directly. Instead, a telephone operator used to sit on the other