Going back to Germany!
A couple of blog posts ago, I wrote that the decision to go to ESC Boston ended up being a great one for many different reasons. I came back from the conference energized and really happy that I went.
These feelings were amplified a few days after my return when I received an email from Rolf Segger, the founder of SEGGER Microcontroller (check out their very new website), asking if I would be interested in visiting their headquarters...
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)
Introduction
This is an article that is a continuation of a digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is recommended that my previous article "Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)"[1] be read first as many sections of this article are directly dependent upon it.
A second family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. It...
Modeling a Continuous-Time System with Matlab
Many of us are familiar with modeling a continuous-time system in the frequency domain using its transfer function H(s) or H(jω). However, finding the time response can be challenging, and traditionally involves finding the inverse Laplace transform of H(s). An alternative way to get both time and frequency responses is to transform H(s) to a discrete-time system H(z) using the impulse-invariant transform [1,2]. This method provides an exact match to the continuous-time...
ESC Boston's Videos are Now Up
In my last blog, I told you about my experience at ESC Boston and the few videos that I was planning to produce and publish. Here they are, please have a look and any feedback (positive or negative) is appreciated.
Short HighlightThis is a very short (one minute) montage of some of the footage that I shot at the show & conference. In future shows, I absolutely need to insert clips here and there of engineers saying a few words about the conference (why they...
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 +...
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 1)
IntroductionThis is an article that is a another digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). Although it is not as far off as the last blog article.
A new family of formulas for calculating the frequency of a single pure tone in a short interval in the time domain is presented. They are a generalization of Equation (1) from Rick Lyons' recent blog article titled "Sinusoidal Frequency Estimation Based on Time-Domain Samples"[1]. ...
Back from ESC Boston
NOT going to ESC Boston would have allowed me to stay home, in my comfort zone.
NOT going to ESC Boston would have saved me from driving in the absolutely horrible & stressful Boston traffic1.
NOT going to ESC Boston would have saved me from having to go through a full search & questioning session at the Canada Customs on my return2.
2017/06/06 update: Videos are now up!So two days...
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...
A Recipe for a Common Logarithm Table
IntroductionThis is an article that is a digression from trying to give a better understanding to the Discrete Fourier Transform (DFT).
A method for building a table of Base 10 Logarithms, also known as Common Logarithms, is featured using math that can be done with paper and pencil. The reader is assumed to have some familiarity with logarithm functions. This material has no dependency on the material in my previous blog articles.
If you were ever curious about how...
Sinusoidal Frequency Estimation Based on Time-Domain Samples
The topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on dsprelated.com. For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact"...
Using the DFT as a Filter: Correcting a Misconception
I have read, in some of the literature of DSP, that when the discrete Fourier transform (DFT) is used as a filter the process of performing a DFT causes an input signal's spectrum to be frequency translated down to zero Hz (DC). I can understand why someone might say that, but I challenge that statement as being incorrect. Here are my thoughts.
Using the DFT as a Filter It may seem strange to think of the DFT as being used as a filter but there are a number of applications where this is...
Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.
This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.
- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:
How Discrete Signal Interpolation Improves D/A Conversion
This blog post is also available in pdf format. Download here.Earlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of dsprelated.com. Here's what I wrote:
We encounter the process of digital-to-analog...
Amplitude modulation and the sampling theorem
I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College. He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.
Most of the code for the chapter is done: you can check it out in this IPython notebook. I haven't written the text yet, but I'll outline it here, and paste in the key figures.
Convolution...
Linear Feedback Shift Registers for the Uninitiated, Part XIII: System Identification
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...
An Efficient Lowpass Filter in Octave
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...
Embedded World 2018 - More Videos!
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...
Bank-switched Farrow resampler
Bank-switched Farrow resampler Summary
A modification of the Farrow structure with reduced computational complexity.Compared to a conventional design, the impulse response is broken into a higher number of segments. Interpolation accuracy is achieved with a lower polynomial order, requiring fewer multiplications per output sample at the expense of a higher overall number of coefficients.
Example codeThis code snippet provides a Matlab / Octave implementation.And
Reduced-Delay IIR Filters
This blog gives the results of a preliminary investigation of reduced-delay (reduced group delay) IIR filters based on my understanding of the concepts presented in a recent interesting blog by Steve Maslen [1].
Development of a Reduced-Delay 2nd-Order IIR Filter
Maslen's development of a reduced-delay 2nd-order IIR filter begins with a traditional prototype filter, HTrad, shown in Figure 1(a). The first modification to the prototype filter is to extract the b0 feedforward coefficient...
Filtering Noise: The Basics (Part 1)
Introduction
Finding signals in the presence of noise is one of the fundamental quests of the discipline of signal processing. Noise is inherently random by nature, so a probability oriented approach is needed to develop a mathematical framework for filtering (i.e. removing/suppressing) noise. This framework or discipline, formally referred to as stochastic signal processing, is often taught in graduate level engineering programs and is covered from different perspectives in excellent...
Errata for the book: 'Understanding Digital Signal Processing'
Errata 3rd Ed. International Version.pdfErrata 3rd Ed. International Version.pdfThis blog post provides, in one place, the errata for each of the many different Editions/Printings of my book Understanding Digital Signal Processing.
If you would like the errata for your copy of the book, merely scroll down and click on the appropriate red line below. For the American versions of the various Editions of the book you'll need to know the "Printing Number" of your copy of the...
Exact Frequency Formula for a Pure Real Tone in a DFT
IntroductionThis 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 major theoretical advance 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...
Back from ESC Boston
NOT going to ESC Boston would have allowed me to stay home, in my comfort zone.
NOT going to ESC Boston would have saved me from driving in the absolutely horrible & stressful Boston traffic1.
NOT going to ESC Boston would have saved me from having to go through a full search & questioning session at the Canada Customs on my return2.
2017/06/06 update: Videos are now up!So two days...
Simple Concepts Explained: Fixed-Point
Introduction
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...
Matlab Code to Synthesize Multiplierless FIR Filters
This article presents Matlab code to synthesize multiplierless Finite Impulse Response (FIR) lowpass filters.
A filter coefficient can be represented as a sum of powers of 2. For example, if a coefficient = decimal 5 multiplies input x, the output is $y= 2^2*x + 2^0*x$. The factor of $2^2$ is then implemented with a shift of 2 bits. This method is not efficient for coefficients having a lot of 1’s, e.g. decimal 31 = 11111. To reduce the number of non-zero...
Multiplierless Exponential Averaging
This blog discusses an interesting approach to exponential averaging. To begin my story, a traditional exponential averager (also called a "leaky integrator"), shown in Figure 1(a), is commonly used to reduce noise fluctuations that contaminate relatively constant-amplitude signal measurements.
Figure 1 Exponential averaging: (a) standard network; (b) single-multiply network.That exponential averager's difference equation is
y(n) = αx(n) + (1 –...Computing Chebyshev Window Sequences
Chebyshev windows (also called Dolph-Chebyshev, or Tchebyschev windows), have several useful properties. Those windows, unlike the fixed Hanning, Hamming, or Blackman window functions, have adjustable sidelobe levels. For a given user-defined sidelobe level and window sequence length, Chebyshev windows yield the most narrow mainlobe compared to any fixed window functions.
However, for some reason, detailed descriptions of how to compute Chebyshev window sequences are not readily available...
A Fast Real-Time Trapezoidal Rule Integrator
This blog presents a computationally-efficient network for computing real‑time discrete integration using the Trapezoidal Rule.
Background
While studying what is called "N-sample Romberg integration" I noticed that such an integration process requires the computation of many individual smaller‑sized integrations using the Trapezoidal Rule integration method [1]. My goal was to create a computationally‑fast real‑time Trapezoidal Rule integration network to increase the processing...
How Not to Reduce DFT Leakage
This blog describes a technique to reduce the effects of spectral leakage when using the discrete Fourier transform (DFT).
In late April 2012 there was a thread on the comp.dsp newsgroup discussing ways to reduce the spectral leakage problem encountered when using the DFT. One post in that thread caught my eye [1]. That post referred to a website presenting a paper describing a DFT leakage method that I'd never heard of before [2]. (Of course, not that I've heard...
Crowdfunding Articles?
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...
