## Second Order Discrete-Time System Demonstration

Discrete-time systems are remarkable: the time response can be computed from mere difference equations, and the coefficients ai, bi of these equations are also the coefficients of H(z). Here, I try to illustrate this remarkableness by converting a continuous-time second-order system to an approximately equivalent discrete-time system. With a discrete-time model, we can then easily compute the time response to any input. But note that the goal here is as much to...

## A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters

This blog discusses the behavior, mathematics, and implementation of cascaded integrator-comb filters.

Cascaded integrator-comb (CIC) digital filters are computationally-efficient implementations of narrowband lowpass filters, and are often embedded in hardware implementations of decimation, interpolation, and delta-sigma converter filtering.

After describing a few applications of CIC filters, this blog introduces their structure and behavior, presents the frequency-domain...

## Are DSPs Dead ?

Are DSPs Dead ?Former Texas Instruments Sr. Fellow Gene Frantz and former TI Fellow Alan Gatherer wrote a 2017 IEEE article about the "death and rebirth" of DSP as a discipline, explaining that now signal processing provides indispensable building blocks in widely popular and lucrative areas such as data science and machine learning. The article implies that DSP will now be taught in university engineering programs as its linear systems and electromagnetics...

## Digging into an Audio Signal and the DSP Process Pipeline

In this post, I'll look at the benefits of using multiple perspectives when handling signals.A Pre-existing Audio FileLet's say we have an audio file of interest. Let's load it into Audacity and zoom in a little (using View → Zoom → Zoom In, multiple times). The figure illustrates the audio signal: just a basic single-tone signal.

By continuing to zoom into the signal, we eventually get to the point of seeing individual samples as illustrated below. Notice that I've marked one...

## A Simplified Matlab Function for Power Spectral Density

In an earlier post [1], I showed how to compute power spectral density (PSD) of a discrete-time signal using the Matlab function pwelch [2]. Pwelch is a useful function because it gives the correct output, and it has the option to average multiple Discrete Fourier Transforms (DFTs). However, a typical function call has five arguments, and it can be hard to remember how to set them all and how they default.

In this post, I create a simplified PSD function by putting a...

## Already 3000+ Attendees Registered for the Upcoming Embedded Online Conference

Chances are you already know, through the newsletter or banners on the Related sites, about the upcoming Embedded Online Conference.

Chances are you also already know that you have until the end of the month of February to register for free.

And chances are that you are one of the more than 3000 pro-active engineers who have already registered.

But If you are like me and have a tendency to do tomorrow what can be done today, maybe you haven't registered yet. You may...

## Fractional Delay FIR Filters

Consider the following Finite Impulse Response (FIR) coefficients:

b = [b0 b1 b2 b1 b0]

These coefficients form a 5-tap symmetrical FIR filter having constant group delay [1,2] over 0 to fs/2 of:

D = (ntaps – 1)/2 = 2 samples

For a symmetrical filter with an odd number of taps, the group delay is always an integer number of samples, while for one with an even number of taps, the group delay is always an integer + 0.5 samples. Can we design a filter...

## The DFT of Finite-Length Time-Reversed Sequences

Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this blog because there is so little coverage of this topic in the literature of DSP.

This blog reviews the two types of time-reversal of finite-length sequences and summarizes their discrete Fourier transform (DFT) frequency-domain characteristics.The Two Types of Time-Reversal in DSP

...## Model Signal Impairments at Complex Baseband

In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise. To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code. Although our example system is QAM, the models can be used for any complex-baseband signal.

I used a very simple complex-baseband model of a QAM system in my last

## Update To: A Wide-Notch Comb Filter

This blog presents alternatives to the wide-notch comb filter described in Reference [1]. That comb filter, which for notational reasons I now call a 2-RRS wide notch comb filter, is shown in Figure 1. I use the "2-RRS" moniker because the comb filter uses two recursive running sum (RRS) networks.

The z-domain transfer function of the 2-RRS wide-notch comb filter, H2-RRS(z), is:

References

[1] R. Lyons, "A Wide-Notch Comb Filter", dsprelated.com Blogs, Nov. 24, 2019, Available...

## An Interesting Fourier Transform - 1/f Noise

Power law functions are common in science and engineering. A surprising property is that the Fourier transform of a power law is also a power law. But this is only the start- there are many interesting features that soon become apparent. This may even be the key to solving an 80-year mystery in physics.

It starts with the following Fourier transform:

The general form is tα ↔ ω-(α+1), where α is a constant. For example, t2 ↔...

## Polyphase Filters and Filterbanks

ALONG CAME POLY

Polyphase filtering is a computationally efficient structure for applying resampling and filtering to a signal. Most digital filters can be applied in a polyphase format, and it is also possible to create efficient resampling filterbanks using the same theories.

This post will walk through a reference implementation of both the downsampling polyphase filter and a downsampling polyphase filterbank using scipy, numpy, matplotlib, and python. It should also highlight some of...

## Plotting Discrete-Time Signals

A discrete-time sinusoid can have frequency up to just shy of half the sample frequency. But if you try to plot the sinusoid, the result is not always recognizable. For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine. But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots. We typically want the plot of a...

## Digital PLL's -- Part 1

1. IntroductionFigure 1.1 is a block diagram of a digital PLL (DPLL). The purpose of the DPLL is to lock the phase of a numerically controlled oscillator (NCO) to a reference signal. The loop includes a phase detector to compute phase error and a loop filter to set loop dynamic performance. The output of the loop filter controls the frequency and phase of the NCO, driving the phase error to zero.

One application of the DPLL is to recover the timing in a digital...

## FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending

This blog presents several interesting things I recently learned regarding the estimation of a spectral value located at a frequency lying between previously computed FFT spectral samples. My curiosity about this FFT interpolation process was triggered by reading a spectrum analysis paper written by three astronomers [1].

My fixation on one equation in that paper led to the creation of this blog.

Background

The notion of FFT interpolation is straightforward to describe. That is, for example,...

## The Exponential Nature of the Complex Unit Circle

IntroductionThis 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...## Simplest Calculation of Half-band Filter Coefficients

Half-band filters are lowpass FIR filters with cut-off frequency of one-quarter of sampling frequency fs and odd symmetry about fs/4 [1]*. And it so happens that almost half of the coefficients are zero. The passband and stopband bandwiths are equal, making these filters useful for decimation-by-2 and interpolation-by-2. Since the zero coefficients make them computationally efficient, these filters are ubiquitous in DSP systems.

Here we will compute half-band...

## Already 3000+ Attendees Registered for the Upcoming Embedded Online Conference

Chances are you already know, through the newsletter or banners on the Related sites, about the upcoming Embedded Online Conference.

Chances are you also already know that you have until the end of the month of February to register for free.

And chances are that you are one of the more than 3000 pro-active engineers who have already registered.

But If you are like me and have a tendency to do tomorrow what can be done today, maybe you haven't registered yet. You may...

## Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes

When the idea of live-streaming parts of Embedded World came to me, I got so excited that I knew I had to make it happen. I perceived the opportunity as a win-win-win-win.

- win #1 - Engineers who could not make it to Embedded World would be able to sample the huge event,
- win #2 - The organisation behind EW would benefit from the extra exposure
- win #3 - Lecturers and vendors who would be live-streamed would reach a (much) larger audience
- win #4 - I would get...

## Generating pink noise

In one of his most famous columns for Scientific American, Martin Gardner wrote about pink noise and its relation to fractal music. The article was based on a 1978 paper by Voss and Clarke, which presents, among other things, a simple algorithm for generating pink noise, also known as 1/f noise.

The fundamental idea of the algorithm is to add up several sequences of uniform random numbers that get updated at different rates. The first source gets updated at...

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

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...

## Delay estimation by FFT

Given x=sig(t) and y=ref(t), returns [c, ref(t+delta), delta)] = fitSignal(y, x);:Estimates and corrects delay and scaling factor between two signals Code snippetThis article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.

IntroductionThere are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...

## Polyphase Filters and Filterbanks

ALONG CAME POLY

Polyphase filtering is a computationally efficient structure for applying resampling and filtering to a signal. Most digital filters can be applied in a polyphase format, and it is also possible to create efficient resampling filterbanks using the same theories.

This post will walk through a reference implementation of both the downsampling polyphase filter and a downsampling polyphase filterbank using scipy, numpy, matplotlib, and python. It should also highlight some of...

## A Beginner's Guide to OFDM

In the recent past, high data rate wireless communications is often considered synonymous to an Orthogonal Frequency Division Multiplexing (OFDM) system. OFDM is a special case of multi-carrier communication as opposed to a conventional single-carrier system.

The concepts on which OFDM is based are so simple that almost everyone in the wireless community is a technical expert in this subject. However, I have always felt an absence of a really simple guide on how OFDM works which can...

## How to Find a Fast Floating-Point atan2 Approximation

Context Over a short period of time, I came across nearly identical approximations of the two parameter arctangent function, atan2, developed by different companies, in different countries, and even in different decades. Fascinated with how the coefficients used in these approximations were derived, I set out to find them. This atan2 implementation is based around a rational approximation of arctangent on the domain -1 to 1:$$ atan(z) \approx \dfrac{z}{1.0 +...

## Music/Audio Signal Processing

Greetings,

This is my blog from the point of view of a music/audio DSP research engineer / educator. It is informal and largely nontechnical because nearly everything I have to say about signal processing is (or will be) somewhere in my four-book series: Mathematics of DFT with Audio Applications, Introduction to Digital Filters, Physical Audio Signal Processing and

## Understanding and Implementing the Sliding DFT

IntroductionIn 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...

## PID Without a PhD

I both consult and teach in the area of digital control. Through both of these efforts, I have found that while there certainly are control problems that require all the expertise I can bring to bear, there are a great number of control problems that can be solved with the most basic knowledge of simple controllers, without resort to any formal control theory at all.

This article will tell you how to implement a simple controller in software and how to tune it without getting into heavy...

## Design IIR Bandpass Filters

In this post, I present a method to design Butterworth IIR bandpass filters. My previous post [1] covered lowpass IIR filter design, and provided a Matlab function to design them. Here, we’ll do the same thing for IIR bandpass filters, with a Matlab function bp_synth.m. Here is an example function call for a bandpass filter based on a 3rd order lowpass prototype:

N= 3; % order of prototype LPF fcenter= 22.5; % Hz center frequency, Hz bw= 5; ...## Embedded Toolbox: Programmer's Calculator

Like any craftsman, I have accumulated quite a few tools during my embedded software development career. Some of them proved to me more useful than others. And these generally useful tools ended up in my Embedded Toolbox. In this blog, I'd like to share some of my tools with you. Today, I'd like to start with my cross-platform Programmer's Calculator called QCalc.

I'm sure that you already have your favorite calculator online or on your smartphone. But can your calculator accept...

## Launch of Youtube Channel: My First Videos - Embedded World 2017

I went to Embedded World 2017 in Nuremberg with an ambitious plan; I would make video highlights of several exhibits (booths) to be presented to the *Related sites audience. I would try to make the vendors focus their pitch on the essential in order to produce a one to three minutes video per booth.

So far my experience with making videos was limited to family videos, so I knew I had lots of reading to do and lots of Youtube videos and tutorials to watch. Trade shows are...

## New Comments System (please help me test it)

I thought it would take me a day or two to implement, it took almost two weeks...

But here it is, the new comments systems for blogs, heavily inspired by the forum system I developed earlier this year.

Which means that:

- You can easily add images, either by drag and drop or through the 'Insert Image' button
- You can add MathML, TeX and ASCIImath equations and they will be rendered with Mathjax
- You can add code snippets and they will be highlighted with highlights.js
- You can edit...

## 3 Good News

Good News #1Last week, I announced a new and ambitious reward program that will be funded by the new Vendors Directory.

This week, I am happy to announce that we have our firsts two sponsors! Quantum Leaps & Abelon Systems have agreed to pay the sponsorship fee to be listed in the new Vendors Directory. Because of their support, there is now some money in the reward pool ($1,000) and enough to pay for the firsts 500 'beers' awarded. Please...

## The New Forum is LIVE!

After months of hard word, I am very excited to introduce to you the new forum interface.

Here are the key features:

1- Easily add images to a post by drag & dropping the images in the editor

2- Easily attach files to a post by drag & dropping the files in the editor

3- Add latex equations to a post and they will be rendered with Mathjax (tutorial)

4- Add a code snippet and surround the code with

## Helping New Bloggers to Break the Ice: A New Ipad Pro for the Author with the Best Article!

Breaking the ice can be tough. Over the years, many individuals have asked to be given access to the blogging interface only to never post an article. Maybe they underestimated the time it takes to write a decent article, or maybe they got cold feet. I don't blame or judge them at all - how many times in my life have I had the intention to do something but didn't follow through? Once, maybe twice 😉 (don't worry if you don't...

## Welcoming MANY New Bloggers!

The response to the latest call for bloggers has been amazing and I am very grateful.

In this post I present to you the individuals who, so far (I am still receiving applications at an impressive rate and will update this page as more bloggers are added), have been given access to the blogging interface. I am very pleased with the positive response and I think the near future will see the publication of many great articles, given the quality of the...

## Recruiting New Bloggers!

Previous calls for bloggers have been very successful in recruiting some great communicators - Rick Lyons, Jason Sachs, Victor Yurkovsky, Mike Silva, Markus Nentwig, Gene Breniman, Stephen Friederichs,

## Premium Forum?

Chances are that by now, you have had a chance to browse the new design of the *related site that I published several weeks ago. I have been working for several months on this and I must admit that I am very happy with the results. This new design will serve as a base for many new exciting developments. I would love to hear your comments/suggestions if you have any, please use the comments system at the bottom of this page.

First on my list would be to build and launch a new forum...

## The Sampling Theorem - An Intuitive Approach

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

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...