Deconvolution by least squares (Using the power of linear algebra in signal processing).

Agustin Bonelli November 12, 20152 comments

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...


The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase

Rick Lyons August 18, 20158 comments

This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the question:

What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?

I'll declare two things to convince you to continue reading.

Declaration# 1: "That the coefficients must be symmetrical" is not a correct


Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1

Cedron Dawg May 21, 2015
Introduction

This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...


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, 2015
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...


The Exponential Nature of the Complex Unit Circle

Cedron Dawg 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...

Sum of Two Equal-Frequency Sinusoids

Rick Lyons September 4, 20141 comment

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...


The wheels go round and round, round and round ...

Christopher Felton May 26, 2014
Introduction

Integer arithmetic is ubiquitous in digital hardware implementations, it's prolific in the control and data-paths.  When using fixed width (constrained) integers, overflow and underflow is business as usual.

Building with Integers

The title of this post mentions a wheel - before I get to the wheel I want to look at an example.  The recursive-windowed-averager (rwa, a.k.a moving average) and its cousin the cascaded-integrator-comb...


DSP Related Math: Nice Animated GIFs

Stephane Boucher April 24, 20143 comments

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...

Instantaneous Frequency Measurement

Parth Vakil February 4, 200820 comments

I would like to talk about the oft used method of measuring the carrier frequency in the world of Signal Collection and Characterization world. It is an elegant technique because of its simplicity. But, of course, with simplicity, there come drawbacks (sometimes...especially with this one!).

In the world of Radar detection and characterization, one of the key characteristics of interest is the carrier frequency of the signal. If the radar is pulsed, you will have a very wide bandwidth, a...


Python scipy.signal IIR Filter Design Cont.

Christopher Felton June 19, 20127 comments

In the previous post the Python scipy.signal iirdesign function was disected.  We reviewed the basics of filter specification and reviewed how to use the iirdesign function to design IIR filters.  The previous post I only demonstrated low pass filter designs.  The following are examples how to use the iirdesign function for highpass, bandpass, and stopband filters designs.

Highpass Filter

The following is a highpass filter design for the different filter...


Sum of Two Equal-Frequency Sinusoids

Rick Lyons September 4, 20141 comment

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...


DSP Related Math: Nice Animated GIFs

Stephane Boucher April 24, 20143 comments

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...

Design of an anti-aliasing filter for a DAC

Markus Nentwig August 18, 2012
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
Problem Figure 1: Typical FIR-DAC-analog lowpass line-up

Digital-to-analog conversion connects digital...


Design study: 1:64 interpolating pulse shaping FIR

Markus Nentwig December 26, 20115 comments

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?”

The Exponential Nature of the Complex Unit Circle

Cedron Dawg 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...

Discrete Wavelet Transform Filter Bank Implementation (part 2)

David Valencia December 5, 20109 comments

Following the previous blog entry: http://www.dsprelated.com/showarticle/115.php

Difference between DWT and DWPT

Before 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 gets...


Signed serial-/parallel multiplication

Markus Nentwig 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...


A brief look at multipath radio channels

Markus Nentwig October 31, 20078 comments

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...