A Fast Guaranteed-Stable Sliding DFT Algorithm

Rick Lyons June 15, 202335 comments

This blog presents a most computationally-efficient guaranteed-stable real-time sliding discrete Fourier transform (SDFT) algorithm. The phrase “real-time” means the network computes one spectral output sample, equal to a single-bin output of an N‑point discrete Fourier transform (DFT), for each input signal sample.

Proposed Guaranteed Stable SDFT

My proposed guaranteed stable SDFT, whose development is given in [1], is shown in Figure 1(a). The output sequence Xk(n) is an N-point...

Return of the Delta-Sigma Modulators, Part 1: Modulation

Jason Sachs May 28, 2023

About a decade ago, I wrote two articles:

Each of these are about delta-sigma modulation, but they’re short and sweet, and not very in-depth. And the 2013 article was really more about analog-to-digital converters. So we’re going to revisit the subject, this time with a lot more technical depth — in fact, I’ve had to split this...

Sonos, Shut Up and Take My Money! - Is Spatial Audio Finally Here?

Stephane Boucher May 24, 20231 comment

Although I generally agree that money can't buy happiness, I recently made a purchase that has brought me countless hours of pure joy. In this blog post, I want to share my excitement with the DSPRelated community, because I know there are many audio and music enthusiasts here, and also because I suspect there is a lot of DSP magic behind this product. And I would love to hear your opinions and experiences if you have also bought or tried the Sonos ERA 300 wireless speaker, or any other...

Simple Discrete-Time Modeling of Lossy LC Filters

Neil Robertson April 19, 2023

There are many software applications that allow modeling LC filters in the frequency domain.  But sometimes it is useful to have a time domain model, such as when you need to analyze a mixed analog and DSP system.  For example, the system in Figure 1 includes an LC filter as well as a DSP portion.  The LC filter could be an anti-alias filter, a channel filter, or some other LC network.  For a design using undersampling, the filter would be bandpass [1].  By modeling...

The Discrete Fourier Transform as a Frequency Response

Neil Robertson February 4, 20236 comments

The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1].  So, if we can find H(k) by whatever method, it should be identical to the DFT of h(n).  In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).

Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one...

Simple Concepts Explained: Fixed-Point

Leandro Stefanazzi January 24, 202311 comments

Most signal processing intensive applications on FPGA are still implemented relying on integer or fixed-point arithmetic. It is not easy to find the key ideas on quantization, fixed-point and integer arithmetic. In a series of articles, I aim to clarify some concepts and add examples on how things are done in real life. The ideas covered are the result of my professional experience and hands-on projects.

In this article I will present the most fundamental question you...

Overview of my Articles

Cedron Dawg December 10, 2022

This article is a summary of all the articles I've written here at DspRelated. The main focus has always been an increased understanding of the Discrete Fourier Transform (DFT). The references are grouped by topic and ordered in a reasonable reading order. All the articles are meant to teach math, or give examples of math, in context within a specific application. Many of the articles also have sample programs which demonstrate the equations derived in the articles. My...

Add the Hilbert Transformer to Your DSP Toolkit, Part 2

Neil Robertson December 4, 20223 comments

In this part, I’ll show how to design a Hilbert Transformer using the coefficients of a half-band filter as a starting point, which turns out to be remarkably simple.  I’ll also show how a half-band filter can be synthesized using the Matlab function firpm, which employs the Parks-McClellan algorithm.

A half-band filter is a type of lowpass, even-symmetric FIR filter having an odd number of taps, with the even-numbered taps (except for the main tap) equal to zero.  This...

Add the Hilbert Transformer to Your DSP Toolkit, Part 1

Neil Robertson November 22, 20224 comments

In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail.  In this article, I’ll dig a little deeper into how the Hilbert Transformer works.  Understanding the Hilbert Transformer involves a modest amount of mathematics, but the payoff in useful applications is worth it.

As we’ll learn, a Hilbert Transformer is just a particular type of Finite Impulse Response (FIR) filter.  In Part 1 of this article, I’ll...

Candan's Tweaks of Jacobsen's Frequency Approximation

Cedron Dawg November 11, 2022

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra calculations beyond the original approximation to come up with an approximation suitable for most...

Computing FFT Twiddle Factors

Rick Lyons August 8, 201019 comments

Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).

Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...

Digital Envelope Detection: The Good, the Bad, and the Ugly

Rick Lyons April 3, 201621 comments

Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.

Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...

Design IIR Butterworth Filters Using 12 Lines of Code

Neil Robertson December 10, 201711 comments

While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch.  You can do it in 12 lines of Matlab code.  In this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order.  Here is an example function call for a 5th order filter:

N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...

Handling Spectral Inversion in Baseband Processing

Eric Jacobsen February 11, 200811 comments

The problem of "spectral inversion" comes up fairly frequently in the context of signal processing for communication systems. In short, "spectral inversion" is the reversal of the orientation of the signal bandwidth with respect to the carrier frequency. Rick Lyons' article on "Spectral Flipping" at http://www.dsprelated.com/showarticle/37.php discusses methods of handling the inversion (as shown in Figure 1a and 1b) at the signal center frequency. Since most communication systems process...

Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm

Rick Lyons July 7, 20151 comment

If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.

Preliminaries To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described by:

$$ Forward \ FFT \rightarrow X(m) = \sum_{n=0}^{N-1} x(n)e^{-j2\pi nm/N} \tag{1} $$ $$ Inverse \ FFT \rightarrow x(n) = {1 \over N} \sum_{m=0}^{N-1}...

Use Matlab Function pwelch to Find Power Spectral Density – or Do It Yourself

Neil Robertson January 13, 201938 comments

In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1].  These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies.  The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the so-called Welch power spectral density estimate [3,4].

In this article, I’ll present some...

The DFT Magnitude of a Real-valued Cosine Sequence

Rick Lyons June 17, 201410 comments

This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

To be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having exactly integer k cycles over N time samples, the peak magnitude of the cosine wave's...

Python scipy.signal IIR Filtering: An Example

Christopher Felton May 19, 2013

In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ).  In this post I am going to conclude the IIR filter design review with an example.

Previous posts:

Understanding and Relating Eb/No, SNR, and other Power Efficiency Metrics

Eric Jacobsen May 29, 20122 comments


Evaluating the performance of communication systems, and wireless systems in particular, usually involves quantifying some performance metric as a function of Signal-to-Noise-Ratio (SNR) or some similar measurement. Many systems require performance evaluation in multipath channels, some in Doppler conditions and other impairments related to mobility. Some have interference metrics to measure against, but nearly all include noise power as an impairment. Not all systems are...

The Exponential Nature of the Complex Unit Circle

Cedron Dawg March 10, 20155 comments

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.


Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World

Stephane Boucher February 21, 2019

Do you have a Twitter and/or Linkedin account?

If you do, please consider paying close attention for the next few days to the EmbeddedRelated Twitter account and to my personal Linkedin account (feel free to connect).  This is where I will be posting lots of updates about how the EmbeddedRelated.tv live streaming experience is going at Embedded World.

The most successful this live broadcasting experience will be, the better the chances that I will be able to do it...

Launch of EmbeddedRelated.tv

Stephane Boucher February 21, 2019

With the upcoming Embedded Word just around the corner, I am very excited to launch the EmbeddedRelated.tv platform.  

This is where you will find the schedule for all the live broadcasts that I will be doing from Embedded World next week.  Please note that the schedule will be evolving constantly, even during the show, so I suggest your refresh the page often.  For instance, I am still unsure if I will be able to do the 'opening of the doors' broadcast as...

Live Streaming from Embedded World!

Stephane Boucher February 12, 2019

For those of you who won't be attending Embedded World this year, I will try to be your eyes and ears by video streaming live from the show floor.   

I am not talking improvised streaming from a phone, but real, high quality HD streaming with a high-end camera and a device that will bond three internet connections (one wifi and two cellular) to ensure a steady, and hopefully reliable, stream. All this to hopefully give those of you who cannot be there in person a virtual...

Sensors Expo - Trip Report & My Best Video Yet!

Stephane Boucher August 3, 20183 comments

This was my first time at Sensors Expo and my second time in Silicon Valley and I must say I had a great time.  

Before I share with you what I find to be, by far, my best 'highlights' video yet for a conference/trade show, let me try to entertain you with a few anecdotes from this trip.  If you are not interested by my stories or maybe don't have the extra minutes needed to read them, please feel free to skip to the end of this blog post to watch the...

Who else is going to Sensors Expo in San Jose? Looking for roommate(s)!

Stephane Boucher May 29, 20186 comments

This will be my first time attending this show and I must say that I am excited. I am bringing with me my cameras and other video equipment with the intention to capture as much footage as possible and produce a (hopefully) fun to watch 'highlights' video. I will also try to film as many demos as possible and share them with you.

I enjoy going to shows like this one as it gives me the opportunity to get out of my home-office (from where I manage and run the *Related sites) and actually...

Crowdfunding Articles?

Stephane Boucher April 12, 201828 comments

Many of you have the knowledge and talent to write technical articles that would benefit the EE community.  What is missing for most of you though, and very understandably so, is the time and motivation to do it.   

But what if you could make some money to compensate for your time spent on writing the article(s)?  Would some of you find the motivation and make the time?

I am thinking of implementing a system/mechanism that would allow the EE community to...

Embedded World 2018 - More Videos!

Stephane Boucher March 27, 20181 comment

After the interview videos last week, this week I am very happy to release two more videos taken at Embedded World 2018 and that I am proud of.  

For both videos, I made extensive use of my two new toys, a Zhiyun Crane Gimbal and a Sony a6300 camera.

The use of a gimbal like the Zhiyun makes a big difference in terms of making the footage look much more stable and cinematographic.

As for the Sony camera, it takes fantastic slow-motion footage and...

Embedded World 2018 - The Interviews

Stephane Boucher March 21, 2018

Once again this year, I had the chance to go to Embedded World in Nuremberg Germany.  And once again this year, I brought my video equipment to try and capture some of the most interesting things at the show.  

Something new this year, I asked Jacob Beningo if he would partner with me in doing interviews with a few vendors.  I would operate the camera while Jacob would ask the right questions to the vendors to make them talk about the key products/features that...

Finally got a drone!

Stephane Boucher August 28, 20172 comments

As a reader of my blog, you already know that I have been making videos lately and thoroughly enjoying the process.  When I was in Germany early this summer (and went 280 km/h in a porsche!) to produce SEGGER's 25th anniversary video, the company bought a drone so we could get an aerial shot of the party (at about the 1:35 mark in this video).  Since then, I have been obsessing on buying a drone for myself and finally made the move a few weeks ago - I acquired a used DJI...