Exact Near Instantaneous Frequency Formulas Best at Zero Crossings
Introduction
This 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]. ...
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"...
Three Bin Exact Frequency Formulas for a Pure Complex Tone in a DFT
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving exact formulas for the frequency of a complex tone in a DFT. This time it is three bin versions. Although the problem is similar to the two bin version in my previous blog article "A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT"[1], a slightly different approach is taken using linear algebra concepts. Because of an extra degree of freedom...
A Two Bin Exact Frequency Formula for a Pure Complex Tone in a DFT
IntroductionThis is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by deriving an exact formula for the frequency of a complex tone in a DFT. It is basically a parallel treatment to the real case given in Exact Frequency Formula for a Pure Real Tone in a DFT. Since a real signal is the sum of two complex signals, the frequency formula for a single complex tone signal is a lot less complicated than for the real case.
Theoretical...DFT Bin Value Formulas for Pure Complex 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 complex tones and an alternative variation. It is basically a parallel treatment to the real case given in DFT Bin Value Formulas for Pure Real Tones. In order to understand how a multiple tone signal acts in a DFT it is necessary to first understand how a single pure tone acts. Since a DFT is a linear transform, the...
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.
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...
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...
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...
Resolving 'Can't initialize target CPU' on TI C6000 DSPs - Part 1
Introduction
Today I am going to discuss some of the basics that can help prevent errors that frustrate some users. The information is directed toward TI C6000 family DSPs, but much of it also applies to other TI DSPs. In many cases they represent the user's first involvement with using Code Composer Studio [CCS] and a target board. It has been my experience that the primary cause of the "Can't initialize target CPU" error message and similar messages like "Error connecting 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...
A Two Bin Solution
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach works best when the tone is between the two DFT bins in terms of frequency.
The Coding...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 ...
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...
Simulink-Simulation of SSB demodulation
≥≥≥ Simulink-Simulation of SSB demodulation or modulation from the article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons Josef Hoffmann
The article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons is a very good description of this topic. The block representation from the figures are clear and easy to understand. They are predestined for a simulation in Simulink. The simulation can help...
Polar Coding Notes: Channel Combining and Channel Splitting
Channel Combining
Channel combining is a step that combines copies of a given B-DMC $W$ in a recursive manner to produce a vector channel $W_N : {\cal X}^N \to {\cal Y}^N$, where $N$ can be any power of two, $N=2^n, n\le0^{[1]}$.
The notation $u_1^N$ as shorthand for denoting a row vector $(u_1, \dots , u_N)$.
The vector channel $W_N$ is the virtual channel between the input sequence $u_1^N$ to a linear encoder and the output sequence $y^N_1$ of $N$...
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?”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...
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...Angle Addition Formulas from Euler's Formula
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT), but only indirectly. The main intent is to get someone who is uncomfortable with complex numbers a little more used to them and relate them back to already known Trigonometric relationships done in Real values. It is essentially a followup to my first blog article "The Exponential Nature of the Complex Unit Circle".
Polar CoordinatesThe more common way of...
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...
Discrete Wavelet Transform Filter Bank Implementation (part 2)
Following the previous blog entry: http://www.dsprelated.com/showarticle/115.php
Difference between DWT and DWPTBefore getting to the equivalent filter obtention, I first want to talk about the difference between DWT(Discrete Wavelet Transform) and DWPT (Discrete Wavelet Packet Transform). The latter is used mostly for image processing.
While DWT has a single "high-pass" branch that filters the signal with the h1 filter, the DWPT separates branches symmetricaly: this means that one...
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...
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...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:
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:
