## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

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 insert L–1...

## 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 NumbersThe variable $ i $ stands for the square root of negative one. This is the...

## Complex Down-Conversion Amplitude Loss

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.")

[This document can be downloaded in pdf format here]

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...

## The Sampling Theorem - An Intuitive Approach

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

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

IntroductionMany 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...

## A Complex Variable Detective Story – A Disconnect Between Theory and Implementation

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) expression by comparing its value, for a given set of filter coefficients, to the...

## The Number 9, Not So Magic After All

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 number 9 [1]. That book is not alone in its fascination with the number 9. If you...

## Sum of Two Equal-Frequency Sinusoids

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 wondered, "Humm ... OK, well then just what are the correct equations for a single...

## The DFT Magnitude of a Real-valued Cosine Sequence

This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

A pdf copy of this blog is availableTo be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having exactly integer k cycles over N time samples, the peak...

## Specifying the Maximum Amplifier Noise When Driving an ADC

I recently learned an interesting rule of thumb regarding the use of an amplifier to drive the input of an analog to digital converter (ADC). The rule of thumb describes how to specify the maximum allowable noise power of the amplifier [1].

A pdf version of this document can be downloaded hereThe Problem Here's the situation for an ADC whose maximum analog input voltage range is –VRef to +VRef. If we drive an ADC's analog input with an sine wave whose peak amplitude is VP = VRef, the ADC's output signal to...

## 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 NumbersThe variable $ i $ stands for the square root of negative one. This is the...

## Free DSP Books on the Internet

While surfing the "net" I have occasionally encountered signal processing books whose chapters could be downloaded to my computer. I started keeping a list of those books and, over the years, that list has grown to over forty books. Perhaps the list will be of interest to you.

Please know, all of the listed books are copyrighted. The copyright holders have graciously provided their books free of charge for downloading for individual use, but multiple copies must not be made or printed. As such, be aware that using any of these books as promotional material is...

## A Fixed-Point Introduction by Example

IntroductionThe finite-word representation of fractional numbers is known as fixed-point. Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation. It extends our finite-word length from a finite set of integers to a finite set of rational real numbers [1]. A fixed-point representation of a number consists of integer and...

## Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

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 insert L–1...

## An Interesting Fourier Transform - 1/f Noise

Power law functions are common in science and engineering. A surprising property is that the Fourier transform of a power law is also a power law. But this is only the start- there are many interesting features that soon become apparent. This may even be the key to solving an 80-year mystery in physics.

It starts with the following Fourier transform:

The general form is tα ↔ ω-(α+1), where α is a constant. For example, t2 ↔ ω–3 and t -0.75 ↔ ω–0.25....

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

Download the pdf versionThere 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...

## Python scipy.signal IIR Filtering: An Example

In 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:

## 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...

## Complex Down-Conversion Amplitude Loss

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.")

[This document can be downloaded in pdf format here]

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...

## Music/Audio Signal Processing

Greetings,

This is my blog from the point of view of a music/audio DSP research engineer / educator. It is informal and largely nontechnical because nearly everything I have to say about signal processing is (or will be) somewhere in my four-book series: Mathematics of DFT with Audio Applications, Introduction to Digital Filters, Physical Audio Signal...

## The Sampling Theorem - An Intuitive Approach

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.

## DSP Related Math: Nice Animated GIFs

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...## DSPRelated and EmbeddedRelated now on Facebook & I will be at EE Live!

I have two news to share with you today.

The first one is that I finally created Facebook pages for DSPRelated.com and EmbeddedRelated (DSPRelated page - EmbeddedRelated page). For a long time I didn't feel that this was something that was needed, but it seems that these days more and more people are using their Facebook account to stay updated with their favorite websites. In any event, if you have a Facebook account, I would greatly appreciate if you...

## 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 several of our guests add to it...

## DSPRelated Finally on Twitter!

Hello!

It's been a while since you've heard from me - and there are many reasons why:

1 - I've made a clown of myself (video here)

2 - I've been working on unifying the user management system. You can now participate to the three related sites (DSPRelated, FPGARelated and EmbeddedRelated) with only one account (same login info).

3- I've been working on getting up to speed with social networks and especially Twitter. I have...

## 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 as a web publisher affords me a lot of flexibility with my schedule, which I am really...

## Do you like the new Comments System?

I have just finished implementing a new comments system for the blogs. Do you like it?

Please share your thoughts with me by adding a comment.

I'll wait a few days and make sure it works properly and then I'll port it to the code snippets and papers section.

Thanks!

## DSP Papers, Articles, Theses, etc

As you may already know, there is a 'Papers and Theses' section on DSPRelated:http://www.dsprelated.com/documents.phpThere are hundreds of DSP Related documents (articles, papers, theses, dissertations, etc) scattered all around the web, and the goal with this section is to find and list as many of those documents as possible in one place. There are, at the moment, a little over 100 documents listed, which I believe is only a small subset of what is available out there, and I need your help to make the list more...

## Code Snippets Suggestions

Despite being only a couple of months old, the Code Snippet section ( DSPRelated.com/code.php ) already contains tens of snippets, thanks to the contributors who have taken the time to share their code.

But let's not stop here - there is room for several hundreds more snippets before the database can be said to cover a decent portion of the DSP field.

To keep the momentum going, I will do two things:

First, I am modifying the rewards program. Instead of paying the rewards based on page...

## Latest DSP Books

As you may already know, Rick Lyons has just published a new edition of his highly acclaimed book: "Understanding Digital Signal Processing". This book has been getting very high ratings and positive reviews from the DSP community since the publication of the first edition. The 3rd edition seems to contain more than enough new material to justify replacing your old copy.

Also of possible interest to you, a new DSP book by C. Britton Rorabaugh...