DFT Bin Value Formulas for Pure Real Tones
Introduction
This 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:
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...
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...
The Exponential Nature of the Complex Unit Circle
IntroductionThis is an article to hopefully give an understanding to Euler's magnificent equation:
$$ e^{i\theta} = cos( \theta ) + i \cdot sin( \theta ) $$
This equation is usually proved using the Taylor series expansion for the given functions, but this approach fails to give an understanding to the equation and the ramification for the behavior of complex numbers. Instead an intuitive approach is taken that culminates in a graphical understanding of the equation.
Complex...Complex Down-Conversion Amplitude Loss
This blog illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")
The general idea behind complex down-conversion is shown in Figure 1(a). And the traditional hardware block diagram of a complex down-converter is shown in Figure 1(b).
Let's assume the input to our down-conversion system is an analog radio frequency (RF) signal,...
The Sampling Theorem - An Intuitive Approach
Scott Kurtz from DSPSoundWare.com has put together a video presentation that aims to help DSPers gain a better intuitive understanding of the Sampling Theorem. Feel free to have a look and share your thoughts by commenting this blog post.
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...
A Complex Variable Detective Story – A Disconnect Between Theory and Implementation
Recently I was in the middle of a pencil-and-paper analysis of a digital 5-tap FIR filter having complex-valued coefficients and I encountered a surprising and thought-provoking problem. So that you can avoid the algebra difficulty I encountered, please read on.
A Surprising Algebra Puzzle
I wanted to derive the H(ω) equation for the frequency response of my FIR digital filter whose complex coefficients were h0, h1, h2, h3, and h4. I could then test the validity of my H(ω)...
The Number 9, Not So Magic After All
This blog is not about signal processing. Rather, it discusses an interesting topic in number theory, the magic of the number 9. As such, this blog is for people who are charmed by the behavior and properties of numbers.
For decades I've thought the number 9 had tricky, almost magical, qualities. Many people feel the same way. I have a book on number theory, whose chapter 8 is titled "Digits — and the Magic of 9", that discusses all sorts of interesting mathematical characteristics of the...
Sum of Two Equal-Frequency Sinusoids
Some time ago I reviewed the manuscript of a book being considered by the IEEE Press publisher for possible publication. In that manuscript the author presented the following equation:
Being unfamiliar with Eq. (1), and being my paranoid self, I wondered if that equation is indeed correct. Not finding a stock trigonometric identity in my favorite math reference book to verify Eq. (1), I modeled both sides of the equation using software. Sure enough, Eq. (1) is not correct. So then I...
Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World
When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...
Deconvolution by least squares (Using the power of linear algebra in signal processing).
When we deal with our normal discrete signal processing operations, like FIR/IIR filtering, convolution, filter design, etc. we normally think of the signals as a constant stream of numbers that we put in a sequence
Errata for the book: 'Understanding Digital Signal Processing'
Errata 3rd Ed. International Version.pdfErrata 3rd Ed. International Version.pdfThis blog post provides, in one place, the errata for each of the many different Editions/Printings of my book Understanding Digital Signal Processing.
If you would like the errata for your copy of the book, merely scroll down and click on the appropriate red line below. For the American versions of the various Editions of the book you'll need to know the "Printing Number" of your copy of the...
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...
Least-squares magic bullets? The Moore-Penrose Pseudoinverse
Hello,
the topic of this brief article is a tool that can be applied to a variety of problems: The Moore-Penrose Pseudoinverse.While maybe not exactly a magic bullet, it gives us least-squares optimal solutions, and that is under many circumstances the best we can reasonably expect.
I'll demonstrate its use on a short example. More details can be found for example on Wikipedia, or the Matlab documentation...
An Efficient Full-Band Sliding DFT Spectrum Analyzer
In this blog I present two computationally efficient full-band discrete Fourier transform (DFT) networks that compute the 0th bin and all the positive-frequency bin outputs for an N-point DFT in real-time on a sample-by-sample basis.
An Even-N Spectrum Analyzer
The full-band sliding DFT (SDFT) spectrum analyzer network, where the DFT size N is an even integer, is shown in Figure 1(a). The x[n] input sequence is restricted to be real-only valued samples. Notice that the only real parts of...
An Astounding Digital Filter Design Application
I've recently encountered a digital filter design application that astonished me with its design flexibility, capability, and ease of use. The software is called the "ASN Filter Designer." After experimenting with a demo version of this filter design software I was so impressed that I simply had publicize it to the subscribers here on dsprelated.com.
What I Liked About the ASN Filter DesignerWith typical filter design software packages the user enters numerical values for the...
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...
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...
Some Observations on Comparing Efficiency in Communication Systems
Introduction
Engineering is usually about managing efficiencies of one sort or another. One of my favorite working definitions of an engineer says, "An engineer is somebody who can do for a nickel what any damn fool can do for a dollar." In that case, the implication is that the cost is one of the characteristics being optimized. But cost isn't always the main efficiency metric, or at least the only one. Consider how a common transportation appliance, the automobile, is optimized...
Impulse Response Approximation
Recently, I stumbled upon a stepped-triangular (ST) approximation that can be implemented as a cascade of recursive running sum (RRS) filters. The following is a short introduction to the stepped-triangular approximation.The stepped-triangular approximation was introduced by Jovanovic-Dolecek and Mitra [1] as a quantized approximation of a low-pass filter (LPF). Figure 1 shows an example of the approximation.
[Figure 1: Stepped Approximation of a LPF...
Interpolator Design: Get the Stopbands Right
In this article, I present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands.
Find Aliased ADC or DAC Harmonics (with animation)
When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics. If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias. In this case, it is not at once obvious if a given spur is a harmonic, and if so, its order. In this article, we’ll present Matlab code to simulate the data converter nonlinearities and find the harmonic alias frequencies. Note that Analog Devices has an online tool for...
There and Back Again: Time of Flight Ranging between Two Wireless Nodes
With the growth in the Internet of Things (IoT) products, the number of applications requiring an estimate of range between two wireless nodes in indoor channels is growing very quickly as well. Therefore, localization is becoming a red hot market today and will remain so in the coming years.
One question that is perplexing is that many companies now a days are offering cm level accurate solutions using RF signals. The conventional wireless nodes usually implement synchronization...
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...
Setting Carrier to Noise Ratio in Simulations
When simulating digital receivers, we often want to check performance with added Gaussian noise. In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N). I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.
Definition of C/NThe Carrier to noise ratio is defined as the ratio of average signal power to noise power for a modulated...
"Neat" Rectangular to Polar Conversion Algorithm
The subject of finding algorithms that estimate the magnitude of a complex number, without having to perform one of those pesky square root operations, has been discussed many times in the past on the comp.dsp newsgroup. That is, given the complex number R + jI in rectangular notation, we want to estimate the magnitude of that number represented by M as:
On August 25th, 2009, Jerry (Mr. Wizard) Avins posted an interesting message on this subject to the comp.dsp newsgroup (Subject: "Re:
A brief look at multipath radio channels
Summary: Discussion of multipath propagation and fading in radio links
Radio channels, their effects on communications links and how to model them are a popular topic on comp.dsp. Unfortunately, for many of us there is little or no opportunity to get any "hands-on" experience with radio-related issues, because the required RF measurement equipment is not that easily available.This article gives a very simple example of a radio link that shows multipath propagation and...
Controlling a DSP Network's Gain: A Note For DSP Beginners
This blog briefly discusses a topic well-known to experienced DSP practitioners but may not be so well-known to DSP beginners. The topic is the proper way to control a digital network's gain. Digital Network Gain Control Figure 1 shows a collection of networks I've seen, in the literature of DSP, where strict gain control is implemented.
FIGURE 1. Examples of digital networks whose initial operations are input signal...
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
