## Finding the Best Optimum

When I was in school learning electrical engineering I owned a large mental pot, full of simmering resentment against the curriculum as it was being taught.

It really started in my junior year, when we took Semiconductor Devices, or more accurately "how to build circuits using transistors". I had been seduced by the pure mathematics of sophomore EE courses, where all the circuit elements (resistors, capacitors, coils and -- oh the joy -- dependent sources) are ideally modeled, and the labs...

## Is It True That *j* is Equal to the Square Root of -1 ?

A few days ago, on the YouTube.com web site, I watched an interesting video concerning complex numbers and the j operator. The video's author claimed that the statement "j is equal to the square root of negative one" is incorrect. What he said was:

He justified his claim by going through the following exercise, starting with:

Based on the algebraic identity:

the author rewrites Eq. (1) as:

If we assume

Eq. (3) can be rewritten...

## Python scipy.signal IIR Filtering: An Example

IntroductionIn the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ). In this post I am going to conclude the IIR filter design review with an example.

Previous posts:

## A Quadrature Signals Tutorial: Complex, But Not Complicated

Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company. Why even Karl Gauss, one the world's greatest mathematicians, called the j-operator the "shadow of...

## FIR sideways (interpolator polyphase decomposition)

An efficient implementation of a symmetric-FIR polyphase 1:3 interpolator that doesn't follow the usual tapped delay line-paradigm. The example exploits the impulse response symmetry and avoids four multiplications out of 10. keywords: symmetric polyphase FIR filter implementation ASIC Matlab / Octave implementation

IntroductionAn interpolating FIR filter can be implemented with a single tapped delay line, possibly going forwards and backwards for a symmetric impulse response. To...

## Design of an anti-aliasing filter for a DAC

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

Digital-to-analog conversion connects digital...

## Understanding the 'Phasing Method' of Single Sideband Demodulation

There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are:

- synchronous detection,
- phasing method,
- Weaver method, and
- filtering method.

Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB signals you might want to scroll down to the 'SSB DEMODULATION BY SYNCHRONOUS DETECTION'...

## Python scipy.signal IIR Filter Design Cont.

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

## Python scipy.signal IIR Filter Design

IntroductionThe following is an introduction on how to design an infinite impulse response (IIR) filters using the Python scipy.signal package. This post, mainly, covers how to use the scipy.signal package and is not a thorough introduction to IIR filter design. For complete coverage of IIR filter design and structure see one of the references.

Filter SpecificationBefore providing some examples lets review the specifications for a filter design. A filter...

## 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...

## 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...

## Implementing Impractical Digital Filters

This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.

Implementing an Impractical Filter: Example 1

Reference [1] presented the digital IIR bandpass filter...

## Feedback Controllers - Making Hardware with Firmware. Part 6. Self-Calibration Related.

This article will consider the engineering of a self-calibration & self-test capability to enable the project hardware to be configured and its basic performance evaluated and verified, ready for the development of the low-latency controller DSP firmware and closed-loop applications. Performance specifications will be documented in due course, on the project website here.

- Part 6: Self-Calibration, Measurements and Signalling (this part)
- Part 5:

## 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...

## Discrete Wavelet Transform Filter Bank Implementation (part 2)

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

Difference between DWT and DWPTBefore 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...

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

Some Design and Simulation Considerations for Sampled-Data ControllersThis 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.

- Part 1: Introduction

## Feedback Controllers - Making Hardware with Firmware. Part 4. Engineering of Evaluation Hardware

Following on from the previous abstract descriptions of an arbitrary circuit emulation application for low-latency feedback controllers, we now come to some aspects in the hardware engineering of an evaluation design from concept to first power-up. In due course a complete specification along with application examples will be maintained on the project website.- Part 1: Introduction
- Part 2:...

## An Alternative Form of the Pure Real Tone DFT Bin Value Formula

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving alternative exact formulas for the bin values of a real tone in a DFT. The derivation of the source equations can be found in my earlier blog article titled "DFT Bin Value Formulas for Pure Real Tones"[1]. The new form is slighty more complicated and calculation intensive, but it is more computationally accurate in the vicinity of near integer frequencies. This...

## Smaller DFTs from bigger DFTs

IntroductionLet's consider the following hypothetical situation: You have a sequence $x$ with $N/2$ points and a black box which can compute the DFT (Discrete Fourier Transform) of an $N$ point sequence. How will you use the black box to compute the $N/2$ point DFT of $x$? While the problem may appear to be a bit contrived, the answer(s) shed light on some basic yet insightful and useful properties of the DFT.

On a related note, the reverse problem of computing an $N$...

## Hidden Linear Algebra in DSP

Linear algebra (LA) is usually thought of as a blunt theoretical subject. However, LA is found hidden in many DSP algorithms used widely in practice.

An obvious clue in finding LA in DSP is the linearity assumption used in theoretical analysis of systems for modelling or design. A standard modelling example for this case would be linear time invariant (LTI) systems. LTI are usually used to model flat wireless communication channels. LTI systems are also used in the design of digital filter...

## 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...

## 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...

## Compute Images/Aliases of CIC Interpolators/Decimators

Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators. For these filters, all coefficients are equal to 1, and there are no multipliers. They are typically used when a large change in sample rate is needed. This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and the aliases of CIC decimators.

1. CIC InterpolatorsFigure 1 shows three interpolate-by-M...

## The Zeroing Sine Family of Window Functions

IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by introducing a class of well behaved window functions that the author believes to be previously unrecognized. The definition and some characteristics are displayed. The heavy math will come in later articles. This is an introduction to the family, and a very special member of it.

This is one of my longer articles. The bulk of the material is in the front half. The...

## Polar Coding Notes: Channel Combining and Channel Splitting

Channel Combining

Channel combining is a step that combines copies of a given B-DMC $W$ in a recursive manner to produce a vector channel $W_N : {\cal X}^N \to {\cal Y}^N$, where $N$ can be any power of two, $N=2^n, n\le0^{[1]}$.

The notation $u_1^N$ as shorthand for denoting a row vector $(u_1, \dots , u_N)$.

The vector channel $W_N$ is the virtual channel between the input sequence $u_1^N$ to a linear encoder and the output sequence $y^N_1$ of $N$...

## Exploring Human Hearing Range

Human Hearing RangeIn this post, I'll look at an interesting aspect of Audacity – using it to explore the threshold of human hearing. In my book Digital Signal Processing: A Gentle Introduction with Audio Examples, I go into this topic and I include a side note on the amazing hearing range of our canine companions.

Creating a Test Audio FileAudacity allows for the generation of a variety of test signals. If you click the Generate->Tone menu, it looks something like...

## [Book Review] Numpy 1.5 Beginner's Guide

Full Disclosure: The publisher of this book, PACKT, was soliciting reviewers. I volunteered to review the book and the publisher sent me an e-version of the book.

IntroductionThe following is a review of "Numpy 1.5 Beginner's Guide", "Learn by doing: less theory, more results" by Ivan Idris. As the title suggests this book is for a beginner. Either someone who is new to numerical computing with high-level languages (HLL) or someone who is new to the Python...

## Implementing a full-duplex UART using the TMS320VC33 serial port

Although the TMS320VC33 serial port was designed to be used as a synchronous port, it can also be used as an asynchronous port under software control. This post describes the hardware and software needed to use a TMS320VC33 serial port as a full-duplex UART port. A schematic diagram and a lengthy code listing are provided to illustrate the solution. This note discusses the implementation of an interrupt-driven, full-duplex, asynchronous serial interface, 9600-baud UART with 8 data bits, 1...

## Unit Testing for Embedded Algorithms

Happy Holidays! For my premier article, I am writing about my favorite technique to use when designing and developing software- unit testing. Unit testing is a best practice when designing software. It allows the designer to verify the behavior of the software units before the entire system is complete, and it facilitates the change and growth of the software system because the developer can verify that the changes will not affect the behavior of other parts of the system. I have used...

## Determination of the transfer function of passive networks with MATLAB Functions

With MATLAB functions, the transfer function of passive networks can be determined relatively easily. The method is explained using the example of a passive low-pass filter of the sixth order, which is shown in Fig.1

Fig.1 Passive low-pass filter of the sixth order

If one tried, as would be logical, to calculate the transfer function starting from the input, it would be quite complicated. On the other hand, if you start from the output, the determination of this function is simple...