## Maximum Likelihood Estimation

Any observation has some degree of noise content that makes our observations uncertain. When we try to make conclusions based on noisy observations, we have to separate the dynamics of a signal from noise. This is the point that estimation starts. Any time that we analyse noisy observations to make decisions, we are estimating some parameters. Parameters are mainly used to simplify the description of a dynamic.

Noise by its definition is a...

## Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering

Recently I've been thinking about digital differentiator and Hilbert transformer implementations and I've developed a processing scheme that may be of interest to the readers here on dsprelated.com.

This blog presents a novel method for simultaneously implementing a digital differentiator (DD), a Hilbert transformer (HT), and a half-band lowpass filter (HBF) using a single tapped-delay line and a single set of coefficients. The method is based on the similarities of the three N =...

## Multimedia Processing with FFMPEG

FFMPEG is a set of libraries and a command line tool for encoding and decoding audio and video in many different formats. It is a free software project for manipulating/processing multimedia data. Many open source media players are based on FFMPEG libraries.

FFMPEG is developed under Linux but it can be compiled under most operating systems including Mac OS, Microsoft Windows. For more details about FFMPEG please refer

## Approximating the area of a chirp by fitting a polynomial

Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs charge).

One way to do that is fitting a curve on our data, and let's face it: this is not that easy. In this post we will work on this issue using Python and its packages. If you do not have Python installed on your system, check here how to...

## 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, such as $x(n)$ with $n\in\mathbb{Z}$. This is at first the most intuitive way of thinking about it, because normally in a digital signal processing system (especially when applied in real time), we take some analogue signal from a sensor like a microphone, convert it...

## Roll Your Own Differentiation Filters

There are many times in digital signal processing that it is necessary to obtain estimates of the derivative of some signal or process from discretely sampled values. Such numerical derivatives are useful in applications such as edge detection, rate of change estimation and optimization, and can even be used as part of quick and efficient dc-blocking filters.

Suppose then, that we want to approximate the derivative of a function at a set of points $\{x_i:i=1,...,N\}$ from...

## Helping New Bloggers to Break the Ice: A New Ipad Pro for the Author with the Best Article!

Breaking the ice can be tough. Over the years, many individuals have asked to be given access to the blogging interface only to never post an article. Maybe they underestimated the time it takes to write a decent article, or maybe they got cold feet. I don't blame or judge them at all - how many times in my life have I had the intention to do something but didn't follow through? Once, maybe twice 😉 (don't worry if you don't...

## GPS - some terminology!

Hi!

For my first post, I will share some information about GPS - Global Positioning System. I will delve one step deeper than a basic explanation of how a GPS system works and introduce some terminology.

GPS, like we all know is the system useful for identifying one's position, velocity, & time using signals from satellites (referred to as SV or space vehicle in literature). It uses the principle of trilateration (not triangulation which is misused frequently) for...

## Welcoming MANY New Bloggers!

The response to the latest call for bloggers has been amazing and I am very grateful.

In this post I present to you the individuals who, so far (I am still receiving applications at an impressive rate and will update this page as more bloggers are added), have been given access to the blogging interface. I am very pleased with the positive response and I think the near future will see the publication of many great articles, given the quality of the...

## Bayes meets Fourier

Joseph Fourier never met Thomas Bayes—Fourier was born in 1768, seven years after Bayes died. But recently I have been exploring connections between the Bayes filter and the Fourier transform.

By "Bayes filter", I don't mean spam filtering using a Bayesian classifier, but rather recursive Bayesian estimation, which is used in robotics and other domains to estimate the state of a system that evolves over time, for example, the position of a moving robot. My interest in...

## Orfanidis Textbooks are Available Online

I have just learned that Sophocles J. Orfanidis, the well-known professor with the ECE Department of Rutgers University, has made two of his signal processing textbooks available for downloading on the Internet. The first textbook is: "Introduction to Signal Processing" available at: http://eceweb1.rutgers.edu/~orfanidi/intro2sp/

Happily, also available at the above web site are:

- Errata for the textbook.
- Homework Solutions Manual
- Errata for Solutions...

## Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm

Someone recently asked me if I knew of a way to compute a fast Fourier transform (FFT) of complex-valued input samples using an FFT algorithm that accepts only real-valued input data. Knowing of no way to do this, I rifled through my library of hardcopy FFT articles looking for help. I found nothing useful that could be applied to this problem.

After some thinking, I believe I have a solution to this problem. Here is my idea:

Let's say our original input data is the complex-valued sequence...

## 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?”## Discrete Wavelet Transform Filter Bank Implementation (part 1)

UPDATE: Added graphs and code to explain the frequency division of the branches

The focus of this article is to briefly explain an implementation of this transform and several filter bank forms. Theoretical information about DWT can be found elsewhere.

First of all, a 'quick and dirty' simplified explanation of the differences between DFT and DWT:

The DWT (Discrete Wavelet Transform), simply put, is an operation that receives a signal as an input (a vector of data) and...

## Peak to Average Power Ratio and CCDF

Peak to Average Power Ratio (PAPR) is often used to characterize digitally modulated signals. One example application is setting the level of the signal in a digital modulator. Knowing PAPR allows setting the average power to a level that is just low enough to minimize clipping.

However, for a random signal, PAPR is a statistical quantity. We have to ask, what is the probability of a given peak power? Then we can decide where to set the average...

## 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

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 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 DFT...

## Benford's law solved with DSP

I have a longtime interest in the mystery of 1/f noise. A few years ago I came across Benford’s law, another puzzle that seemed to have many of the same characteristics.

Suppose you collect a large group of seemingly random numbers, such as might appear in a newspaper or financial report. Benford’s law relates to the leading digit of each number, such as "4" in 4.268, "3" in 0.0312, and "9" in -932.34. Since there are nine possible leading digits...

## Computing the Group Delay of a Filter

I just learned a new method (new to me at least) for computing the group delay of digital filters. In the event this process turns out to be interesting to my readers, this blog describes the method. Let's start with a bit of algebra so that you'll know I'm not making all of this up.

Assume we have the N-sample h(n) impulse response of a digital filter, with n being our time-domain index, and that we represent the filter's discrete-time Fourier transform (DTFT), H(ω), in polar form...

## 3 Good News

Good News #1Last week, I announced a new and ambitious reward program that will be funded by the new Vendors Directory.

This week, I am happy to announce that we have our firsts two sponsors! Quantum Leaps & Abelon Systems have agreed to pay the sponsorship fee to be listed in the new Vendors Directory. Because of their support, there is now some money in the reward pool ($1,000) and enough to pay for the firsts 500 'beers' awarded. Please...

## Spline interpolation

A cookbook recipe for segmented y=f(x) 3rd-order polynomial interpolation based on arbitrary input data. Includes Octave/Matlab design script and Verilog implementation example. Keywords: Spline, interpolation, function modeling, fixed point approximation, data fitting, Matlab, RTL, Verilog

IntroductionSplines describe a smooth function with a small number of parameters. They are well-known for example from vector drawing programs, or to define a "natural" movement path through given...

## Signed serial-/parallel multiplication

Keywords: Binary signed multiplication implementation, RTL, Verilog, algorithm

Summary- A detailed discussion of bit-level trickstery in signed-signed multiplication
- Algorithm based on Wikipedia example
- Includes a Verilog implementation with parametrized bit width

A straightforward method to multiply two binary numbers is to repeatedly shift the first argument a, and add to a register if the corresponding bit in the other argument b is set. The...

## Benford's law solved with DSP

I have a longtime interest in the mystery of 1/f noise. A few years ago I came across Benford’s law, another puzzle that seemed to have many of the same characteristics.

Suppose you collect a large group of seemingly random numbers, such as might appear in a newspaper or financial report. Benford’s law relates to the leading digit of each number, such as "4" in 4.268, "3" in 0.0312, and "9" in -932.34. Since there are nine possible leading digits...

## Two jobs

For those of you following closely embeddedrelated and the other related sites, you might have noticed that I have been less active for the last couple of months, and I will use this blog post to explain why. The main reason is that I got myself involved into a project that ended up using a better part of my cpu than I originally thought it would.

I currently have two jobs: one as an electrical/dsp engineer recycled as a web publisher and the other as a parent of three kids. My job...

## Collaborative Writing Experiment: Your Favorite DSP Websites

You are invited to contribute to the content of this blog post through the magic of Google Docs' real time collaboration feature.

I discovered this tool several months ago when I was looking for a way to coordinate our annual family halloween party (potluck) and avoid the very unpleasant situation of ending up with too much chips and not enough chocolate (first world problem!). It was amusing to keep an eye on the "food you will bring" document we had created for this and watch...

## Using Mason's Rule to Analyze DSP Networks

There have been times when I wanted to determine the z-domain transfer function of some discrete network, but my algebra skills failed me. Some time ago I learned Mason's Rule, which helped me solve my problems. If you're willing to learn the steps in using Mason's Rule, it has the power of George Foreman's right hand in solving network analysis problems.

This blog discusses a valuable analysis method (well known to our analog control system engineering brethren) to obtain the z-domain...

## Setting the 3-dB Cutoff Frequency of an Exponential Averager

This blog discusses two ways to determine an exponential averager's weighting factor so that the averager has a given 3-dB cutoff frequency. Here we assume the reader is familiar with exponential averaging lowpass filters, also called a "leaky integrators", to reduce noise fluctuations that contaminate constant-amplitude signal measurements. Exponential averagers are useful because they allow us to implement lowpass filtering at a low computational workload per output sample.

Figure 1 shows...

## A multiuser waterfilling algorithm

Hello,this blog entry documents a code snippet for a multi-user waterfilling algorithm. It's heuristic and relatively straightforward, making it easy to implement additional constraints or rules.I rewrote parts of it to improve readability, but no extensive testing took place afterwards. Please double-check that it does what it promises.

Introduction to multiuser waterfilling.Background information can be found for example in the presentation from Yosia Hadisusanto,

## The Little Fruit Market

There used to be a fruit market located at 391 San Antonio Road in Mountain View, California. In the 1990's I worked part time in Mountain View and drove past this market's building, shown in Figure 1, many times, unaware of its history. What happened at that fruit market has changed the lives of almost everyone on our planet. Here's the story.

William Shockley In 1948 the brilliant physicist William Shockley, along with John Bardeen and Walter Brattain, co-invented the transistor at Bell...

## 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...