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...
An s-Plane to z-Plane Mapping Example
While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.
Reader, please take a few moments to see if you detect any errors in Figure 1.
...Should DSP Undergraduate Students Study z-Transform Regions of Convergence?
Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:
"I noticed that you didn't discuss z-plane regions of convergence here. In my undergraduate DSP class we spent a lot of classroom and homework time on the ...
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...
Recruiting New Bloggers!
Previous calls for bloggers have been very successful in recruiting some great communicators - Rick Lyons, Jason Sachs, Victor Yurkovsky, Mike Silva, Markus Nentwig, Gene Breniman, Stephen Friederichs,
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...
Half-band filter on Xilinx FPGA
1. DSP48 Slice in Xilinx FPGA
There are many DSP48 Slices in most Xilinx® FPGAs, one DSP48 slice in Spartan6® FPGA is shown in Figure 1, the structure may different depending on the device, but broadly similar.
Figure 1: A whole DSP48A1 Slice in Spartan6 (www.xilinx.com)
2. Symmetric Systolic Half-band FIRFigure 2: Symmetric Systolic Half-band FIR Filter
3. Two-channel Symmetric Systolic Half-band FIRFigure 3: 2-Channel...
Time-Domain Periodicity and the Discrete Fourier Transform
Introduction
The Discrete Fourier Transform (DFT) and it's fast-algorithm implementation, the Fast Fourier Transform (FFT), are fundamental tools for processing and analysis of digital signals. While the continuous Fourier Transform and its inverse integrate over all time from minus infinity to plus infinity, and all frequencies from minus infinity to plus infinity, practical application of its discrete cousins can only be made over finite time and frequency intervals. The discrete nature...
A Remarkable Bit of DFT Trivia
I recently noticed a rather peculiar example of discrete Fourier transform (DFT) trivia; an unexpected coincidence regarding the scalloping loss of the DFT. Here's the story.
DFT SCALLOPING LOSS As you know, if we perform an N-point DFT on N real-valued time-domain samples of a discrete sine wave, whose frequency is an integer multiple of fs/N (fs is the sample rate in Hz), the peak magnitude of the sine wave's positive-frequency spectral component will be
where A is the peak amplitude...
Ancient History
The other day I was downloading an IDE for a new (to me) OS. When I went to compile some sample code, it failed. I went onto a forum, where I was told "if you read the release notes you'd know that the peripheral libraries are in a legacy download". Well damn! Looking back at my previous versions I realized I must have done that and forgotten about it. Everything changes, and keeping up with it takes time and effort.
When I first started with microprocessors we...
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 2
Configuration
The previous article discussed CCS configuration. The prerequisite for the following discussion is a valid CCS configuration file. All references will be for CCS 3.3, but they may be used or adapted to other versions of CCS. From the previous discussion, we know that the configuration file is located at 'C:\CCStudio_v3.3\cc\bin\brddat\ccBrd0.dat'.
XDS510 Emulators
Initial discussion will address only XDS510 class emulators that support TI drivers and utilities. This will...
Waveforms that are their own Fourier Transform
Mea Culpa
There are many scary things about writing a technical book. Can I make the concepts clear? It is worth the effort? Will it sell? But all of these pale compared to the biggest fear: What if I'm just plain wrong? Not being able to help someone is one thing, but leading them astray is far worse.
My book on DSP has now been published for almost ten years. I've found lots of typos, a few misstatements, and many places where the explanations confuse even me. But I have been lucky;...
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 Intel numerically controlled oscillator
Understanding Radio Frequency Distortion
Overview
The topic of this article are the effects of radio frequency distortions on a baseband signal, and how to model them at baseband. Typical applications are use as a simulation model or in digital predistortion algorithms.
IntroductionTransmitting and receiving wireless signals usually involves analog radio frequency circuits, such as power amplifiers in a transmitter or low-noise amplifiers in a receiver.Signal distortion in those circuits deteriorates the link quality. When...
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...
The Discrete Fourier Transform as a Frequency Response
The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1]. So, if we can find H(k) by whatever method, it should be identical to the DFT of h(n). In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n).
Consider the four-tap FIR filter in Figure 1, where each block labeled Ts represents a delay of one...
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...
Specifying the Maximum Amplifier Noise When Driving an ADC
I recently learned an interesting rule of thumb regarding the use of an amplifier to drive the input of an analog to digital converter (ADC). The rule of thumb describes how to specify the maximum allowable noise power of the amplifier [1].
The Problem Here's the situation for an ADC whose maximum analog input voltage range is –VRef to +VRef. If we drive an ADC's analog input with an sine wave whose peak amplitude is VP = VRef, the ADC's output signal to noise ratio is maximized. We'll...
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 2
Configuration
The previous article discussed CCS configuration. The prerequisite for the following discussion is a valid CCS configuration file. All references will be for CCS 3.3, but they may be used or adapted to other versions of CCS. From the previous discussion, we know that the configuration file is located at 'C:\CCStudio_v3.3\cc\bin\brddat\ccBrd0.dat'.
XDS510 Emulators
Initial discussion will address only XDS510 class emulators that support TI drivers and utilities. This will...
Adaptive Beamforming is like Squeezing a Water Balloon
Adaptive beamforming was first developed in the 1960s for radar and sonar applications. The main idea is that signals can be captured using multiple sensors and the sensor outputs can be combined to enhance the signals propagating from specific directions and attenuate (null out) signals from other directions. It has grown immensely in recent years as processors have become faster and cheaper. Today, adaptive beamforming applications include smart speakers (like the Amazon Echo),...
Sonos, Shut Up and Take My Money! - Is Spatial Audio Finally Here?
Although I generally agree that money can't buy happiness, I recently made a purchase that has brought me countless hours of pure joy. In this blog post, I want to share my excitement with the DSPRelated community, because I know there are many audio and music enthusiasts here, and also because I suspect there is a lot of DSP magic behind this product. And I would love to hear your opinions and experiences if you have also bought or tried the Sonos ERA 300 wireless speaker, or any other...
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...
A Simpler Goertzel Algorithm
In this blog I propose a Goertzel algorithm that is simpler than the version of the Goertzel algorithm that is traditionally presented DSP textbooks. Below I very briefly describe the DSP textbook version of the Goertzel algorithm followed by a description of my proposed simpler algorithm.
The Traditional DSP Textbook Goertzel Algorithm
The so-called Goertzel algorithm is used to efficiently compute a single mth-bin sample of an N-point discrete Fourier transform (DFT) [1-4]. The...
Harmonic Notch Filter
My basement is covered with power lines and florescent lights which makes collecting ECG and EEG data rather difficult due to the 60 cycle hum. I found the following notch filter to work very well at eliminating the background signal without effecting the highly amplified signals I was looking for.
The notch filter is based on the a transfer function with the form $$H(z)=\frac{1}{2}(1+A(z))$$ where A(z) is an all pass filter. The original paper [1] describes a method to...
