New Video: Parametric Oscillations
I just posted this last night. It's kinda off-topic from the mission of the channel, but I realized that it had been months since I'd posted a video, and having an excuse to build on helped keep me on track.
Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine
Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.
One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...
Some Thoughts on Sampling
Some time ago, I came across an interesting problem. In the explanation of sampling process, a representation of impulse sampling shown in Figure 1 below is illustrated in almost every textbook on DSP and communications. The question is: how is it possible that during sampling, the frequency axis gets scaled by $1/T_s$ -- a very large number? For an ADC operating at 10 MHz for example, the amplitude of the desired spectrum and spectral replicas is $10^7$! I thought that there must be...
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...
Wavelets II - Vanishing Moments and Spectral Factorization
In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.
Besides the content of the last post, you should be familiar with basic complex algebra, the...
Fibonacci trick
I'm working on a video, tying the Fibonacci sequence into the general subject of difference equations.
Here's a fun trick: take any two consecutive numbers in the Fibonacci sequence, say 34 and 55. Now negate one and use them as the seed for the Fibonacci sequence, larger magnitude first, i.e.
$-55, 34, \cdots$
Carry it out, and you'll eventually get the Fibonacci sequence, or it's negative:
$-55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1 \cdots$
This is NOT a general property of difference...
The Power Spectrum
Often, when calculating the spectrum of a sampled signal, we are interested in relative powers, and we don’t care about the absolute accuracy of the y axis. However, when the sampled signal represents an analog signal, we sometimes need an accurate picture of the analog signal’s power in the frequency domain. This post shows how to calculate an accurate power spectrum.
Parseval’s theorem [1,2] is a property of the Discrete Fourier Transform (DFT) that...
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...
Wavelets I - From Filter Banks to the Dilation Equation
This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...
The Real Star of Star Trek
Unless you've been living under a rock recently, you're probably aware that this month is the 50-year anniversary of the original Star Trek show on American television. It's an anniversary worth noting, as did Time and Newsweek magazines with their special editions.
Over the years I've come to realize that a major star of the original Star Trek series wasn't an actor. It was a thing. The starship USS Enterprise! Before I explain my thinking, here's a little...
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 –...Linear Feedback Shift Registers for the Uninitiated, Part XIV: Gold Codes
Last time we looked at some techniques using LFSR output for system identification, making use of the peculiar autocorrelation properties of pseudorandom bit sequences (PRBS) derived from an LFSR.
This time we’re going to jump back to the field of communications, to look at an invention called Gold codes and why a single maximum-length PRBS isn’t enough to save the world using spread-spectrum technology. We have to cover two little side discussions before we can get into Gold...
Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images
This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.
Background
The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.
Figure 1
The '↑ L' operation in Figure 1 means to...
Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved
This blog clarifies how to obtain and interpret the z-domain transfer function of the coupled-form 2nd-order IIR resonator. The coupled-form 2nd-order IIR resonator was developed to overcome a shortcoming in the standard 2nd-order IIR resonator. With that thought in mind, let's take a brief look at a standard 2nd-order IIR resonator.
Standard 2nd-Order IIR Resonator A block diagram of the standard 2nd-order IIR resonator is shown in Figure 1(a). You've probably seen that block diagram many...
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...
A poor man's Simulink
Glue between Octave and NGSPICE for discrete- and continuous time cosimulation (download) Keywords: Octave, SPICE, Simulink
IntroductionMany DSP problems have close ties with the analog world. For example, a switched-mode audio power amplifier uses a digital control loop to open and close power transistors driving an analog filter. There are commercial tools for digital-analog cosimulation: Simulink comes to mind, and mainstream EDA vendors support VHDL-AMS or Verilog-A in their...
Number Theory for Codes
Everything in the digital world is encoded. ASCII and Unicode are combinations of bits which have specific meanings to us. If we try to interpret a compiled program as Unicode, the result is a lot of garbage (and beeps!) To reduce errors in transmissions over radio links we use Error Correction Codes so that even when bits are lost we can recover the ASCII or Unicode original. To prevent anyone from understanding a transmission we can encrypt the raw data...
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"...
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
...Wavelets I - From Filter Banks to the Dilation Equation
This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...
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...
Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm
Someone recently asked me if I knew of a way to compute a fast Fourier transform (FFT) of complex-valued input samples using an FFT algorithm that accepts only real-valued input data. Knowing of no way to do this, I rifled through my library of hardcopy FFT articles looking for help. I found nothing useful that could be applied to this problem.
After some thinking, I believe I have a solution to this problem. Here is my idea:
Let's say our original input data is the complex-valued sequence...
SEGGER's 25th Anniversary Video
Chances are you will find this video more interesting to watch if you take five minutes to first read the story of the week I spent at SEGGER's headquarters at the end of June.
The video is only a little more than 2 minutes long. If you decide to watch it, make sure to go full screen and I would really love to read your thoughts about it in the comments down bellow. Do you think a video like this succeeds in making the viewer want to learn more about the company?...
Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals
In digital signal processing (DSP) we're all familiar with the processes of bandpass sampling an analog bandpass signal and downsampling a digital bandpass signal. The overall spectral behavior of those operations are well-documented. However, mathematical expressions for computing the translated frequency of individual spectral components, after bandpass sampling or downsampling, are not available in the standard DSP textbooks. The following three sections explain how to compute the...
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...
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
Introduction to the topic
This is the 1st in a series of articles looking at how we can use DSP and Feedback Control Sciences along with some mixed-signal electronics and number-crunching capability (e.g. FPGA), to create arbitrary (within reason) Electrical/Electronic Circuits with real-world connectivity. Of equal importance will be the evaluation of the functionality and performance of a practical design made from modestly-priced state of the art devices.
- Part 1:
Summary of ROC Rules
This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.
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...
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...
A Brief Introduction To Romberg Integration
This blog briefly describes a remarkable integration algorithm, called "Romberg integration." The algorithm is used in the field of numerical analysis but it's not so well-known in the world of DSP.
To show the power of Romberg integration, and to convince you to continue reading, consider the notion of estimating the area under the continuous x(t) = sin(t) curve based on the five x(n) samples represented by the dots in Figure 1.The results of performing a Trapezoidal Rule, a...
