## Constrained Integer Behavior

The wheels go round and round, round and round ...Integer arithmetic is ubiquitous in digital hardware implementations, it's prolific in the control and data-paths. When using fixed width (constrained) integers, overflow and underflow is business as usual.

Building with IntegersThe subtitle of this post mentions a wheel - before I get to the wheel I want to look at an example. The recursive-windowed-averager (rwa, a.k.a moving average)...

## 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...## Signed serial-/parallel multiplication

Keywords: Binary signed multiplication implementation, RTL, Verilog, algorithm

Summary- A detailed discussion of bit-level trickstery in signed-signed multiplication
- Algorithm based on Wikipedia example
- Includes a Verilog implementation with parametrized bit width

A straightforward method to multiply two binary numbers is to repeatedly shift the first argument a, and add to a register if the corresponding bit in the other argument b is set. The...

## Finding the Best Optimum

When I was in school learning electrical engineering I owned a large mental pot, full of simmering resentment against the curriculum as it was being taught.

It really started in my junior year, when we took Semiconductor Devices, or more accurately "how to build circuits using transistors". I had been seduced by the pure mathematics of sophomore EE courses, where all the circuit elements (resistors, capacitors, coils and -- oh the joy -- dependent sources) are ideally modeled, and the labs...

## Is It True That *j* is Equal to the Square Root of -1 ?

A few days ago, on the YouTube.com web site, I watched an interesting video concerning complex numbers and the j operator. The video's author claimed that the statement "j is equal to the square root of negative one" is incorrect. What he said was:

He justified his claim by going through the following exercise, starting with:

Based on the algebraic identity:

the author rewrites Eq. (1) as:

If we assume

Eq. (3) can be rewritten...

## Python scipy.signal IIR Filtering: An Example

IntroductionIn 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:

## A Quadrature Signals Tutorial: Complex, But Not Complicated

Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company. Why even Karl Gauss, one the world's greatest mathematicians, called the j-operator the "shadow of...

## FIR sideways (interpolator polyphase decomposition)

An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation

IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...

## Design of an anti-aliasing filter for a DAC

Overview- Octaveforge / Matlab design script. Download: here
- weighted numerical optimization of Laplace-domain transfer function
- linear-phase design, optimizes vector error (magnitude and phase)
- design process calculates and corrects group delay internally
- includes sinc() response of the sample-and-hold stage in the ADC
- optionally includes multiplierless FIR filter

Digital-to-analog conversion connects digital...

## Understanding the 'Phasing Method' of Single Sideband Demodulation

There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are:

- synchronous detection,
- phasing method,
- Weaver method, and
- filtering method.

Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB signals you might want to scroll down to the 'SSB DEMODULATION BY SYNCHRONOUS DETECTION'...

## Instant CIC

Summary:

A floating point model for a CIC decimator, including the frequency response.

Description:

A CIC filter relies on a peculiarity of its fixed-point implementation: Normal operation involves repeated internal overflows that have no effect to the output signal, as they cancel in the following stage.

One way to put it intuitively is that only the speed (and rate of change) of every little "wheel" in the clockworks carries information, but its absolute position is...

## Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects

Some Design and Simulation Considerations for Sampled-Data ControllersThis article will continue to look at some aspects of the controllers and electronics needed to create emulated physical circuits with real-world connectivity and will look at the issues that arise in sampled-data controllers compared to continuous-domain controllers. As such, is not intended as an introduction to sampled-data systems.

- Part 1: Introduction

## Feedback Controllers - Making Hardware with Firmware. Part 4. Engineering of Evaluation Hardware

Following on from the previous abstract descriptions of an arbitrary circuit emulation application for low-latency feedback controllers, we now come to some aspects in the hardware engineering of an evaluation design from concept to first power-up. In due course a complete specification along with application examples will be maintained on the project website.- Part 1: Introduction
- Part 2:...

## FIR sideways (interpolator polyphase decomposition)

An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation

IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...

## There's No End to It -- Matlab Code Plots Frequency Response above the Unit Circle

Reference [1] has some 3D plots of frequency response magnitude above the unit circle in the Z-plane. I liked them enough that I wrote a Matlab function to plot the response of any digital filter this way. I’m not sure how useful these plots are, but they’re fun to look at. The Matlab code is listed in the Appendix.This post is available in PDF format for easy...

## Python number crunching faster? Part I

Everyone has their favorite computing platform, regardless if it is Matlab, Octave, Scilab, Mathematica, Mathcad, etc. I have been using Python and the common numerical and scientific packages available. Personally, I have found this to be very useful in my work. Lately there has been some chatter on speeding up Python.

From another project I follow, MyHDL, I was introduced to the Python JIT compiler,

## Exponential Smoothing with a Wrinkle

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.

Basic Exponential Smoothing

Exponential smoothing is also known as...

## Implementing Impractical Digital Filters

This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.

Implementing an Impractical Filter: Example 1

Reference [1] presented the digital IIR bandpass filter...

## Two Bin Exact Frequency Formulas 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 exact formulas for the frequency of a real tone in a DFT. This time it is a two bin version. The approach taken is a vector based one similar to the approach used in "Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT"[1]. The real valued formula presented in this article actually preceded, and was the basis for the complex three bin...

## 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:

## DFT Graphical Interpretation: Centroids of Weighted Roots of Unity

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by framing it in a graphical interpretation. The bin calculation formula is shown to be the equivalent of finding the center of mass, or centroid, of a set of points. Various examples are graphed to illustrate the well known properties of DFT bin values. This treatment will only consider real valued signals. Complex valued signals can be analyzed in a similar manner with...

## DFT Bin Value Formulas for Pure Real Tones

IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an analytical formula for the DFT of pure real tones. The formula is used to explain the well known properties of the DFT. A sample program is included, with its output, to numerically demonstrate the veracity of the formula. This article builds on the ideas developed in my previous two blog articles:

## Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka

For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2]. In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file. Then, we can take the CSV file and open it in Weka. Once in Weka, we have a lot of paths to consider in order to classify it. I use the term classify loosely since there are many things you can do with data sets...

## [Book Review] Numpy 1.5 Beginner's Guide

Full Disclosure: The publisher of this book, PACKT, was soliciting reviewers. I volunteered to review the book and the publisher sent me an e-version of the book.

IntroductionThe following is a review of "Numpy 1.5 Beginner's Guide", "Learn by doing: less theory, more results" by Ivan Idris. As the title suggests this book is for a beginner. Either someone who is new to numerical computing with high-level languages (HLL) or someone who is new to the Python...

## Unit Testing for Embedded Algorithms

Happy Holidays! For my premier article, I am writing about my favorite technique to use when designing and developing software- unit testing. Unit testing is a best practice when designing software. It allows the designer to verify the behavior of the software units before the entire system is complete, and it facilitates the change and growth of the software system because the developer can verify that the changes will not affect the behavior of other parts of the system. I have used...

## Filter a Rectangular Pulse with no Ringing

To filter a rectangular pulse without any ringing, there is only one requirement on the filter coefficients: they must all be positive. However, if we want the leading and trailing edge of the pulse to be symmetrical, then the coefficients must be symmetrical. What we are describing is basically a window function.

Consider a rectangular pulse 32 samples long with fs = 1 kHz. Here is the Matlab code to generate the pulse:

N= 64; fs= 1000; % Hz sample...## Feedback Controllers - Making Hardware with Firmware. Part 2. Ideal Model Examples

Developing and Validating Simulation ModelsThis article will describe models for simulating the systems and controllers for the hardware emulation application described in Part 1 of the series.

- Part 1: Introduction
- Part 2: Ideal Model Examples
- Part 3: Sampled Data Aspects
- Part 4: Engineering of Evaluation Hardware
- Part 5:

## 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...

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

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...

## Implementing a full-duplex UART using the TMS320VC33 serial port

Although the TMS320VC33 serial port was designed to be used as a synchronous port, it can also be used as an asynchronous port under software control. This post describes the hardware and software needed to use a TMS320VC33 serial port as a full-duplex UART port. A schematic diagram and a lengthy code listing are provided to illustrate the solution. This note discusses the implementation of an interrupt-driven, full-duplex, asynchronous serial interface, 9600-baud UART with 8 data bits, 1...