## Exponential Smoothing with a Wrinkle

December 17, 2015
Introduction

This is an article to hopefully give a better understanding to the Discrete Fourier Transform (DFT) by providing a set of preprocessing filters to improve the resolution of the DFT. Because of the exponential nature of sinusoidal functions, they have special mathematical properties when exponential smoothing is applied to them. These properties are derived and explained in this blog article.

Basic Exponential Smoothing

Exponential smoothing is also known as...

## Differentiating and integrating discrete signals

December 14, 20152 comments

I am back at work on Think DSP, adding a new chapter on differentiation and integration.  In the previous chapter (which you can read here) I present Gaussian smoothing, show how smoothing in the time domain corresponds to a low-pass filter in the frequency domain, and present the Convolution Theorem.

In the current chapter, I start with the first difference operation (diff in Numpy) and show that it corresponds to a high-pass filter in the frequency domain.  I use historical stock...

## Discrete-Time PLLs, Part 1: Basics

December 1, 20153 comments

Design Files: Part1.slx

Hi everyone,

In this series of tutorials on discrete-time PLLs we will be focusing on Phase-Locked Loops that can be implemented in discrete-time signal proessors such as FPGAs, DSPs and of course, MATLAB.

In the first part of the series, we will be reviewing the basics of continuous-time baseband PLLs and we will see some useful mathematics that will give us insight into the inners working of PLLs. In the second part, we will focus on...

## 60 numbers

November 30, 20152 comments

This blog title is inspired from the Peabody award-winning Radiolab episode 60 words. Radiolab is well known for its insightful stories on Science with an amazing sound design. Today's blog is about decoding Radiolab's theme music (actually, just a small "Mmm Newewe" part of it hereafter called the Radiolab sound). I have been taking this online course on Audio Signal Processing where we are taught how to analyze sounds...

## Compressive Sensing - Recovery of Sparse Signals (Part 1)

November 29, 2015

The amount of data that is generated has been increasing at a substantial rate since the beginning of the digital revolution. The constraints on the sampling and reconstruction of digital signals are derived from the well-known Nyquist-Shannon sampling theorem. To review, the theorem states that a band-limited signal, with the highest frequency of $f_{max}$, can be completely reconstructed from its samples if the sampling rate, $f_{s}$, is at least twice the signal bandwidth. If the...

## Summary of ROC Rules

November 26, 20152 comments

This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.

Summary of ROC RulesFor a causal system the ROC extends outwards.For a non-causal system the ROC extends inwards.For a two-sided system, the ROC can extend inwards or outwards from every pole. The ROC cannot contain any polesThe system is stable if the unity circle is included in the ROCOne Pole System...

## Analytic Signal

November 26, 20154 comments

In communication theory and modulation theory we always deal with two phases: In-phase (I) and Quadrature-phase (Q). The question that I will discuss in this blog is that why we use two phases and not more.

Any real band-limited signal along with its Hilbert transformed pair form an analytic signal. We normally use the analytic signal for modulation. A modulated signal is actually a carrier or the sine signal that one attribute of it is changing with time which is our signal....

## Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka

November 25, 2015

For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2].  In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.  Then, we can take the CSV file and open it in Weka.  Once in Weka, we have a lot of paths to consider in order to classify it.  I use the term classify loosely since there are many things you can do with data sets...

## Maximum Likelihood Estimation

November 24, 2015

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

November 24, 20152 comments

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 =...

## Signed serial-/parallel multiplication

February 16, 2014

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
Signed serial-/parallel multiplication

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...

## Two jobs

December 5, 201223 comments

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...

## Spline interpolation

May 11, 20142 comments

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

Introduction

Splines 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...

## Benford's law solved with DSP

February 23, 20087 comments

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...

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

October 22, 20126 comments

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...

## Go Big or Go Home - Generating $500,000 for Contributors February 18, 20168 comments In a Nutshell • A new Vendors Directory has been created • Vendors will be invited to pay a sponsorship fee to be listed in the directory • 100% of the basic sponsorship fee will be distributed to the *Related Sites community through a novel reward system • The goal is for the directory to eventually generate - drum roll please -$500,000 on a yearly basis for contributing members on the *Related Sites
• Members will choose how the reward money gets distributed between...

## Using Mason's Rule to Analyze DSP Networks

August 31, 20096 comments

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...

## Optimizing the Half-band Filters in Multistage Decimation and Interpolation

January 4, 201615 comments

This blog discusses a not so well-known rule regarding the filtering in multistage decimation and interpolation by an integer power of two. I'm referring to sample rate change systems using half-band lowpass filters (LPFs) as shown in Figure 1. Here's the story.

Figure 1: Multistage decimation and interpolation using half-band filters.

Multistage Decimation – A Very Brief Review

Figure 2(a) depicts the process of decimation by an integer factor D. That...

## A poor man's Simulink

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...

## The Exponential Nature of the Complex Unit Circle

March 10, 20152 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...