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

March 9, 20206 comments
In this post, I'll look at the benefits of using multiple perspectives when handling signals.A Pre-existing Audio File

Let'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

March 3, 20204 comments

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

February 14, 2020

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

February 9, 202012 comments

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

December 20, 201910 comments

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

December 11, 20192 comments

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

December 9, 2019

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...

## A Wide-Notch Comb Filter

November 24, 201918 comments

This blog describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.

Background

Let's first review the behavior of a traditional comb filter. Figure 1(a) shows a traditional comb filter comprising two cascaded recursive running sum (RRS) comb filters. Figure 1(b) shows the filter's co-located dual poles and dual zeros on the z-plane, while Figure 1(c) shows the filter's positive-frequency magnitude response when, for example, D = 9. The...

## An Efficient Lowpass Filter in Octave

November 6, 2019

This article describes an efficient linear-phase lowpass FIR filter, coded using the Octave programming language. The intention is to focus on the implementation in software, but references are provided for those who wish to undertake further study of interpolated FIR filters [1]- [3].

The input signal is processed as a vector of samples (eg from a .wav file), which are converted to a matrix format.   The complete filter is thus referred to as a Matrix IFIR or...

## Compute Modulation Error Ratio (MER) for QAM

November 5, 20192 comments

This post defines the Modulation Error Ratio (MER) for QAM signals, and shows how to compute it.  As we’ll see, in the absence of impairments other than noise, the MER tracks the signal’s Carrier-to-Noise Ratio (over a limited range).  A Matlab script at the end of the PDF version of this post computes MER for a simplified QAM-64 system.

Figure 1 is a simplified block diagram of a QAM system.  The transmitter includes a source of QAM symbols, a root-Nyquist...

## Delay estimation by FFT

September 22, 200750 comments
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 snippet

This article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.

Introduction

There 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

March 19, 201310 comments

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...

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

January 13, 201933 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].

## Music/Audio Signal Processing

September 5, 20087 comments

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

## A Beginner's Guide to OFDM

May 1, 20176 comments

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

May 26, 201714 comments
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 +...

## Understanding and Implementing the Sliding DFT

April 23, 201511 comments
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...

## Embedded Toolbox: Programmer's Calculator

June 27, 20178 comments

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...

## Pulse Shaping in Single-Carrier Communication Systems

April 10, 200833 comments

Some common conceptual hurdles for beginning communications engineers have to do with "Pulse Shaping" or the closely-related, even synonymous, topics of "matched filtering", "Nyquist filtering", "Nyquist pulse", "pulse filtering", "spectral shaping", etc. Some of the confusion comes from the use of terms like "matched filter" which has a broader meaning in the more general field of signal processing or detection theory. Likewise "Raised Cosine" has a different meaning or application in this...

## PID Without a PhD

April 26, 201611 comments

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...

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

April 5, 201721 comments

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...

October 4, 201618 comments

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

March 9, 20161 comment
Good News #1

Last 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!

February 18, 20161 comment

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!

November 9, 2015

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!

October 27, 20153 comments

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!

October 16, 20157 comments

Previous calls for bloggers have been very successful in recruiting some great communicators - Rick LyonsJason Sachs, Victor Yurkovsky, Mike Silva, Markus NentwigGene BrenimanStephen Friederichs,

May 25, 201514 comments

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

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, 20143 comments

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...