Beat frequency sinusoid and its spectrum

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

Continue reading
DTMF spectrum for key 6

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

Continue reading
A DNA double helix

The Concept of Frequency

A wireless signal from one device to another travels through the use of electromagnetic waves propagated by an antenna. Electromagnetic waves have different frequencies and one can pick up a specific signal by tuning a radio Rx to a specific frequency. But what is a frequency anyway? Watch the video below for an interesting description of actual time domain samples and how to interpret their frequency domain representation. A Complex Sinusoid Consider a complex number $V$ in an $IQ$-plane. A complex number is defined as a pair of real numbers in $(x,y)$-plane similar to the vectors but with different arithmetic

Continue reading
A complex number with its I and Q components

Dealing with Complex Numbers

Although complex notation is not complex to understand, I attempt to avoid complex notation altogether while writing DSP articles. If you are interested in where these complex numbers come from, you can read my real-imaginative guide to complex numbers. A complex number is defined as an ordered pair of real numbers in $(x,y)$-plane. In that respect, complex numbers can be considered as vectors with initial point on the origin $(0,0)$. Addition of complex numbers is then similar to the addition of vectors in $(x,y)$-plane from this perspective. However, multiplication is well defined for complex numbers while it is not defined

Continue reading