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Exact Frequency Formula for a Pure Real Tone in a DFT

Cedron Dawg April 20, 20152 comments
Introduction

This 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...


DFT Bin Value Formulas for Pure Real Tones

Cedron Dawg April 17, 20151 comment
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

Cedron Dawg April 10, 20151 comment
Introduction

This 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

Rick Lyons March 23, 20154 comments

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

Cedron Dawg March 10, 20155 comments
Introduction

This 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

Rick Lyons March 3, 20157 comments

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

Stephane Boucher January 26, 20151 comment

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

Markus Nentwig January 24, 20153 comments

Glue between Octave and NGSPICE for discrete- and continuous time cosimulation (download) Keywords: Octave, SPICE, Simulink

Introduction

Many 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

Rick Lyons October 14, 2014

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

Rick Lyons October 1, 20146 comments

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...


DAC Zero-Order Hold Models

Neil Robertson January 21, 20242 comments

This article provides two simple time-domain models of a DAC’s zero-order hold. These models will allow us to find time and frequency domain approximations of DAC outputs, and simulate analog filtering of those outputs. Developing the models is also a good way to learn about the DAC ZOH function.


Third-Order Distortion of a Digitally-Modulated Signal

Neil Robertson June 9, 2020
Analog designers are always harping about amplifier third-order distortion.  Why?  In this article, we’ll look at why third-order distortion is important, and simulate a QAM signal with third-order distortion.

In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude.  With this assumption, the output y of a non-ideal amplifier can be written as a power series of the input signal x:

$$y=...


Feedback Controllers - Making Hardware with Firmware. Part 5. Some FPGA Aspects.

Steve Maslen November 14, 2017
This part of the on-going series of articles looks at a variety of aspects concerning the FPGA device which provides the high-speed maths capability for the low-latency controller and the arbitrary circuit generator application. In due course a complete specification along with  application  examples will be maintained on the project website here.

Feedback Controllers - Making Hardware with Firmware. Part 3. Sampled Data Aspects

Steve Maslen September 9, 2017
Some Design and Simulation Considerations for Sampled-Data Controllers

This 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.


An Efficient Full-Band Sliding DFT Spectrum Analyzer

Rick Lyons April 1, 20217 comments

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...


Make Hardware Great Again

Jeff Brower June 29, 20205 comments

By now you're aware of the collective angst in the US about 5G. Why is the US not a leader in 5G ? Could that also happen -- indeed, is it happening -- in AI ? If we lead in other areas, why not 5G ? What makes it so hard ?

This hand-wringing has reached the highest levels in US government. Recently the Wall Street Journal reported on a DoJ promoted plan 1 to help Cisco buy Ericsson or Nokia, to give the US a leg up in 5G. This is not a new plan,...


Modelling a Noisy Communication Signal in MATLAB for the Analog to Digital Conversion Process

Parth Vakil October 30, 200713 comments

A critical thing to realize while modeling the signal that is going to be digitally processed is the SNR. In a receiver, the noise floor (hence the noise variance and hence its power) are determined by the temperature and the Bandwidth. For a system with a constant bandwidth, relatively constant temperature, the noise power remains relatively constant as well. This implies that the noise variance is a constant.

In MATLAB, the easiest way to create a noisy signal is by using...


There and Back Again: Time of Flight Ranging between Two Wireless Nodes

Qasim Chaudhari October 23, 20175 comments

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...


Ancient History

Mike January 18, 20168 comments

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...


The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance

Rick Lyons September 24, 201933 comments

This blog shows the danger in evaluating the performance of a digital integration network based solely on its frequency response curve. If you plan on implementing a digital integrator in your signal processing work I recommend you continue reading this blog.

Background

Typically when DSP practitioners want to predict the accuracy performance of a digital integrator they compare how closely that integrator's frequency response matches the frequency response of an ideal integrator [1,2]....