Improved Three Bin Exact Frequency Formula for a Pure Real Tone in a DFT
Introduction
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by extending the exact two bin formulas for the frequency of a real tone in a DFT to the three bin case. This article is a direct extension of my prior article "Two Bin Exact Frequency Formulas for a Pure Real Tone in a DFT"[1]. The formulas derived in the previous article are also presented in this article in the computational order, rather than the indirect order they were...
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...
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:...
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 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 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:
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
Introduction to the topicThis 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:
Exact Near Instantaneous Frequency Formulas Best at Zero Crossings
IntroductionThis is an article that is the last of my digression from trying to give a better understanding of the Discrete Fourier Transform (DFT). It is along the lines of the last two.
In those articles, I presented exact formulas for calculating the frequency of a pure tone signal as instantaneously as possible in the time domain. Although the formulas work for both real and complex signals (something that does not happen with frequency domain formulas), for real signals they...
Exact Near Instantaneous Frequency Formulas Best at Peaks (Part 2)
IntroductionThis 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...
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]. ...
The DFT Output and Its Dimensions
The Discrete Fourier Transform, or DFT, converts a signal from discrete time to discrete frequency. It is commonly implemented as and used as the Fast Fourier Transform (FFT). This article will attempt to clarify the format of the DFT output and how it is produced.
Living in the real world, we deal with real signals. The data we typically sample does not have an imaginary component. For example, the voltage sampled by a receiver is a real value at a particular point in time. Let’s...
Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...
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...
Dealing With Fixed Point Fractions
Fixed point fractional representation always gives me a headache because I screw it up the first time I try to implement an algorithm. The difference between integer operations and fractional operations is in the overflow. If the representation fits in the fixed point result, you can not tell the difference between fixed point integer and fixed point fractions. When integers overflow, they lose data off the most significant bits. When fractions overflow, they lose data off...
Feedback Controllers - Making Hardware with Firmware. Part I. Introduction
Introduction to the topicThis 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:
Candan's Tweaks of Jacobsen's Frequency Approximation
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by explaining how a tweak to a well known frequency approximation formula makes it better, and another tweak makes it exact. The first tweak is shown to be the first of a pattern and a novel approximation formula is made from the second. It only requires a few extra calculations beyond the original approximation to come up with an approximation suitable for most...
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
Feedback Controllers - Making Hardware with Firmware. Part 8. Control Loop Test-bed
This part in the series will consider the signals, measurements, analyses and configurations for testing high-speed low-latency feedback loops and their controllers. Along with basic test signals, a versatile IFFT signal generation scheme will be discussed and implemented. A simple controller under test will be constructed to demonstrate the analysis principles in preparation for the design and evaluation of specific controllers and closed-loop applications.
Additional design...Phase and Amplitude Calculation for a Pure Complex Tone in a DFT using Multiple Bins
IntroductionThis 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...
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
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...
Design study: 1:64 interpolating pulse shaping FIR
This article is the documentation to a code snippet that originated from a discussion on comp.dsp.
The task is to design a root-raised cosine filter with a rolloff of a=0.15 that interpolates to 64x the symbol rate at the input.
The code snippet shows a solution that is relatively straightforward to design and achieves reasonably good efficiency using only FIR filters.
Motivation: “simple solutions?”Curse you, iPython Notebook!
First, I think ipython is great. I use it daily and always have an ipython terminal open. But just recently, I was showing off the ipython 0.12 notebook and in the process created a lengthy example while demonstrating the cool features of the ipython notebook. The example included LaTeX equations, plots, etc. Since the notebook session was on something of relevance I decided to clean up the session and use it for the beginning of a report.
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...
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...
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...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...
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:
How precise is my measurement?
Some might argue that measurement is a blend of skepticism and faith. While time constraints might make you lean toward faith, some healthy engineering skepticism should bring you back to statistics. This article reviews some practical statistics that can help you satisfy one common question posed by skeptical engineers: “How precise is my measurement?” As we’ll see, by understanding how to answer it, you gain a degree of control over your measurement time.
An accurate, precise...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...
