Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins

Cedron Dawg March 14, 2018
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from several DFT bin values and knowing the frequency. This article is functionally an extension of my prior article "Phase and Amplitude Calculation for a Pure Complex Tone in a DFT"[1] which used only one bin for a complex tone, but it is actually much more similar to my approach for real...


Linear Feedback Shift Registers for the Uninitiated, Part XIII: System Identification

Jason Sachs March 12, 20181 comment

Last time we looked at spread-spectrum techniques using the output bit sequence of an LFSR as a pseudorandom bit sequence (PRBS). The main benefit we explored was increasing signal-to-noise ratio (SNR) relative to other disturbance signals in a communication system.

This time we’re going to use a PRBS from LFSR output to do something completely different: system identification. We’ll show two different methods of active system identification, one using sine waves and the other...


Coefficients of Cascaded Discrete-Time Systems

Neil Robertson March 4, 2018

In this article, we’ll show how to compute the coefficients that result when you cascade discrete-time systems.  With the coefficients in hand, it’s then easy to compute the time or frequency response.  The computation presented here can also be used to find coefficients of mixed discrete-time and continuous-time systems, by using a discrete time model of the continuous-time portion [1].

This article is available in PDF format for...


Design IIR Filters Using Cascaded Biquads

Neil Robertson February 11, 2018
This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads.  We’ll derive how to calculate the coefficients of the biquads and do some examples using a Matlab function biquad_synth provided in the Appendix.  Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter (Chebyshev, Bessel, etc).  As we’ll see, the cascaded-biquad design is less sensitive to coefficient...

Design IIR Highpass Filters

Neil Robertson February 3, 2018

This post is the fourth in a series of tutorials on IIR Butterworth filter design.  So far we covered lowpass [1], bandpass [2], and band-reject [3] filters; now we’ll design highpass filters.  The general approach, as before, has six steps:

Find the poles of a lowpass analog prototype filter with Ωc = 1 rad/s. Given the -3 dB frequency of the digital highpass filter, find the corresponding frequency of the analog highpass filter (pre-warping). Transform the...

Design IIR Band-Reject Filters

Neil Robertson January 17, 2018

In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis.  Earlier posts covered IIR Butterworth lowpass [1] and bandpass [2] filters.  Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB frequencies.   I’ll discuss the differences between the two approaches later in this...


Design IIR Bandpass Filters

Neil Robertson January 6, 20184 comments

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; ...

Phase and Amplitude Calculation for a Pure Complex Tone in a DFT

Cedron Dawg January 6, 2018
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas to calculate the phase and amplitude of a pure complex tone from a DFT bin value and knowing the frequency. This is a much simpler problem to solve than the corresponding case for a pure real tone which I covered in an earlier blog article[1]. In the noiseless single tone case, these equations will be exact. In the presence of noise or other tones...


Feedback Controllers - Making Hardware with Firmware. Part 7. Turbo-charged DSP Oscillators

Steve Maslen January 5, 20187 comments
This article will look at some DSP Sine-wave oscillators and will show how an FPGA with limited floating-point performance due to latency, can be persuaded to produce much higher sample-rate sine-waves of high quality. 

Comparisons will be made between implementations on Intel Cyclone V and Cyclone 10 GX FPGAs. An...


Linear Feedback Shift Registers for the Uninitiated, Part XII: Spread-Spectrum Fundamentals

Jason Sachs December 29, 20171 comment

Last time we looked at the use of LFSRs for pseudorandom number generation, or PRNG, and saw two things:

  • the use of LFSR state for PRNG has undesirable serial correlation and frequency-domain properties
  • the use of single bits of LFSR output has good frequency-domain properties, and its autocorrelation values are so close to zero that they are actually better than a statistically random bit stream

The unusually-good correlation properties...


Python scipy.signal IIR Filter Design

Christopher Felton May 14, 20124 comments
Introduction

The following is an introduction on how to design an infinite impulse response (IIR) filters using the Python scipy.signal package.  This post, mainly, covers how to use the scipy.signal package and is not a thorough introduction to IIR filter design.  For complete coverage of IIR filter design and structure see one of the references.

Filter Specification

Before providing some examples lets review the specifications for a filter design.  A filter...


Embedded Toolbox: Programmer's Calculator

Miro Samek 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...


Delay estimation by FFT

Markus Nentwig September 22, 200745 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,...


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

Rick Lyons April 3, 20168 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)...


The DFT Magnitude of a Real-valued Cosine Sequence

Rick Lyons June 17, 20148 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...


Sum of Two Equal-Frequency Sinusoids

Rick Lyons September 4, 20142 comments

Some time ago I reviewed the manuscript of a book being considered by the IEEE Press publisher for possible publication. In that manuscript the author presented the following equation:

Being unfamiliar with Eq. (1), and being my paranoid self, I wondered if that equation is indeed correct. Not finding a stock trigonometric identity in my favorite math reference book to verify Eq. (1), I modeled both sides of the equation using software. Sure enough, Eq. (1) is not correct. So then I...


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}...

Music/Audio Signal Processing

Julius Orion Smith III 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


Understanding and Preventing Overflow (I Had Too Much to Add Last Night)

Jason Sachs December 4, 2013

Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.

In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:

for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...

PID Without a PhD

Tim Wescott 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...


New Comments System (please help me test it)

Stephane Boucher 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

Stephane Boucher 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...


Go Big or Go Home - Generating $500,000 for Contributors

Stephane Boucher February 18, 20168 comments
In a Nutshell
  • A new Vendors Directory has been created
  • Vendors will be invited to pay a sponsorship fee to be listed in the directory
  • 100% of the basic sponsorship fee will be distributed to the *Related Sites community through a novel reward system
  • The goal is for the directory to eventually generate - drum roll please -  $500,000 on a yearly basis for contributing members on the *Related Sites
  • Members will choose how the reward money gets distributed between...

The New Forum is LIVE!

Stephane Boucher 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!

Stephane Boucher 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!

Stephane Boucher 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!

Stephane Boucher 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,


Premium Forum?

Stephane Boucher 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

Stephane Boucher 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

Stephane Boucher 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...