## Understanding and Implementing the Sliding DFT

April 23, 2015
Introduction

In many applications the detection or processing of signals in the frequency domain offers an advantage over performing the same task in the time-domain.   Sometimes the advantage is just a simpler or more conceptually straightforward algorithm, and often the largest barrier to working in the frequency domain is the complexity or latency involved in the Fast Fourier Transform computation.   If the frequency-domain data must be updated frequently in a real-time application, the complexity and latency of the FFT can become a significant impediment to...

## Exact Frequency Formula for a Pure Real Tone in a DFT

April 20, 2015
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a real tone in a DFT. According to current teaching, this is not possible, so this article should be considered a watershed event in the discipline. The formula is presented in a few different formats. Some sample calculations are provided to give a numerical demonstration of the formula in use. This article is an extension of my previous blog article:

## DFT Bin Value Formulas for Pure Real Tones

April 17, 2015
Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure real tones. The formula is used to explain the well known properties of the DFT. A sample program is included, with its output, to numerically demonstrate the veracity of the formula. This article builds on the ideas developed in my previous two blog articles:

## DFT Graphical Interpretation: Centroids of Weighted Roots of Unity

April 10, 2015
Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will only consider real valued signals. Complex valued signals can be analyzed in a similar manner with the only distinction being rotation as well as rescaling will occur. Most of the signals analyzed...

## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

March 23, 2015

This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.

Background

The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.

Figure 1

The '↑ L' operation in Figure 1 means to insert L–1...

## The Exponential Nature of the Complex Unit Circle

Introduction

This is an article to hopefully give an understanding to Euler's magnificent equation:

$$e^{i\theta} = cos( \theta ) + i \cdot sin( \theta )$$

This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.

Complex Numbers

The variable $i$ stands for the square root of negative one. This is the...

## Complex Down-Conversion Amplitude Loss

March 3, 20151 comment

This blog illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")

The general idea behind complex down-conversion is shown in Figure 1(a). And the traditional hardware block diagram of a complex down-converter is shown in Figure 1(b).

Let's assume the...

## The Sampling Theorem - An Intuitive Approach

January 26, 20151 comment

Scott Kurtz from DSPSoundWare.com has put together a video presentation that aims to help DSPers gain a better intuitive understanding of the Sampling Theorem.   Feel free to have a look and share your thoughts by commenting this blog post.

Introduction

Many DSP problems have close ties with the analog world. For example, a switched-mode audio power amplifier uses a digital control loop to open and close power transistors driving an analog filter. There are commercial tools for digital-analog cosimulation: Simulink comes to mind, and mainstream EDA vendors support VHDL-AMS or...

## A Complex Variable Detective Story – A Disconnect Between Theory and Implementation

October 14, 2014

Recently I was in the middle of a pencil-and-paper analysis of a digital 5-tap FIR filter having complex-valued coefficients and I encountered a surprising and thought-provoking problem. So that you can avoid the algebra difficulty I encountered, please read on.

A Surprising Algebra Puzzle

I wanted to derive the H(ω) equation for the frequency response of my FIR digital filter whose complex coefficients were h0, h1, h2, h3, and h4. I could then test the validity of my H(ω) expression by comparing its value, for a given set of filter coefficients, to the...

## The Sampling Theorem - An Intuitive Approach

January 26, 20151 comment

Scott Kurtz from DSPSoundWare.com has put together a video presentation that aims to help DSPers gain a better intuitive understanding of the Sampling Theorem.   Feel free to have a look and share your thoughts by commenting this blog post.

## DSP Related Math: Nice Animated GIFs

April 24, 20141 comment

I was browsing the ECE subreddit lately and found that some of the most popular posts over the last few months have been animated GIFs helping understand some mathematical concepts.  I thought there would be some value in aggregating the DSP related gifs on one page.

The relationship between sin, cos, and right triangles: Constructing a square wave with infinite series (see this...

## DSPRelated and EmbeddedRelated now on Facebook & I will be at EE Live!

I have two news to share with you today.

The first one is that I finally created Facebook pages for DSPRelated.com and EmbeddedRelated (DSPRelated page - EmbeddedRelated page). For a long time I didn't feel that this was something that was needed, but it seems that these days more and more people are using their Facebook account to stay updated with their favorite websites. In any event, if you have a Facebook account, I would greatly appreciate if you...

## Collaborative Writing Experiment: Your Favorite DSP Websites

May 30, 2013

You are invited to contribute to the content of this blog post through the magic of Google Docs' real time collaboration feature.

I discovered this tool several months ago when I was looking for a way to coordinate our annual family halloween party (potluck) and avoid the very unpleasant situation of ending up with too much chips and not enough chocolate (first world problem!).  It was amusing to keep an eye on the "food you will bring" document we had created for this and watch several of our guests add to it...

Hello!

It's been a while since you've heard from me - and there are many reasons why:

2 - I've been working on unifying the user management system.  You can now participate to the three related sites (DSPRelated, FPGARelated and EmbeddedRelated) with only one account (same login info).

3- I've been working on getting up to speed with social networks and especially Twitter.   I have...

## Two jobs

For those of you following closely embeddedrelated and the other related sites, you might have noticed that I have been less active for the last couple of months, and I will use this blog post to explain why. The main reason is that I got myself involved into a project that ended up using a better part of my cpu than I originally thought it would.

I currently have two jobs: one as an electrical/dsp engineer recycled as a web publisher and the other as a parent of three kids. My job as a web publisher affords me a lot of flexibility with my schedule, which I am really...

## Do you like the new Comments System?

I have just finished implementing a new comments system for the blogs.  Do you like it?

I'll wait a few days and make sure it works properly and then I'll port it to the code snippets and papers section.

Thanks!

## DSP Papers, Articles, Theses, etc

March 17, 20111 comment

As you may already know, there is a 'Papers and Theses' section on DSPRelated:http://www.dsprelated.com/documents.phpThere are hundreds of DSP Related documents (articles, papers, theses, dissertations, etc) scattered all around the web, and the goal with this section is to find and list as many of those documents as possible in one place. There are, at the moment, a little over 100 documents listed, which I believe is only a small subset of what is available out there, and I need your help to make the list more...

## Code Snippets Suggestions

Despite being only a couple of months old, the Code Snippet section ( DSPRelated.com/code.php ) already contains tens of snippets, thanks to the contributors who have taken the time to share their code.

But let's not stop here - there is room for several hundreds more snippets before the database can be said to cover a decent portion of the DSP field.

To keep the momentum going, I will do two things:

First, I am modifying the rewards program.  Instead of paying the rewards based on page...

## Latest DSP Books

December 1, 2010

As you may already know, Rick Lyons has just published a new edition of his highly acclaimed book: "Understanding Digital Signal Processing".   This book has been getting very high ratings and positive reviews from the DSP community since the publication of the first edition.  The 3rd edition seems to contain more than enough new material to justify replacing your old copy.

Also of possible interest to you, a new DSP book by C. Britton Rorabaugh...