## Find Aliased ADC or DAC Harmonics (with animation)

When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics. If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias. In this case, it is not at once obvious if a given spur is a harmonic, and if so, its order. In this article, we’ll present Matlab code to simulate the data converter nonlinearities and find the harmonic alias frequencies. Note that Analog Devices has an online tool for...

## Adaptive Beamforming is like Squeezing a Water Balloon

Adaptive beamforming was first developed in the 1960s for radar and sonar applications. The main idea is that signals can be captured using multiple sensors and the sensor outputs can be combined to enhance the signals propagating from specific directions and attenuate (null out) signals from other directions. It has grown immensely in recent years as processors have become faster and cheaper. Today, adaptive beamforming applications include smart speakers (like the Amazon Echo),...

## 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...

## 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...

## The DSP Online Conference - Right Around the Corner!

It is Sunday night as I write this blog post with a few days to go before the virtual doors of the very first DSP Online Conference open..

It all started with a post in the DSPRelated forum about three months ago. We had just had a blast running the 2020 Embedded Online Conference and we thought it could be fun to organize a smaller event dedicated to the DSP community. So my goal with the post in the forum was to see if...

## 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...

## Design Square-Root Nyquist Filters

In his book on multirate signal processing, harris presents a nifty technique for designing square-root Nyquist FIR filters with good stopband attenuation [1]. In this post, I describe the method and provide a Matlab function for designing the filters. You can find a Matlab function by harris for designing the filters at [2].

BackgroundSingle-carrier modulation, such as QAM, uses filters to limit the bandwidth of the signal. Figure 1 shows a simplified QAM system block...

## Make Hardware Great Again

By now you're aware of the collective angst in the US about 5G. Why is the US not a leader in 5G ? Could that also happen -- indeed, is it happening -- in AI ? If we lead in other areas, why not 5G ? What makes it so hard ?

This hand-wringing has reached the highest levels in US government. Recently the Wall Street Journal reported on a DoJ promoted plan 1 to help Cisco buy Ericsson or Nokia, to give the US a leg up in 5G. This is not a new plan,...

## A Fast Real-Time Trapezoidal Rule Integrator

This blog presents a computationally-efficient network for computing real‑time discrete integration using the Trapezoidal Rule.

Background

While studying what is called "N-sample Romberg integration" I noticed that such an integration process requires the computation of many individual smaller‑sized integrations using the Trapezoidal Rule integration method [1]. My goal was to create a computationally‑fast real‑time Trapezoidal Rule integration network to increase the processing...

## Third-Order Distortion of a Digitally-Modulated Signal

Analog designers are always harping about amplifier third-order distortion. Why? In this article, we’ll look at why third-order distortion is important, and simulate a QAM signal with third-order distortion.

In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude. With this assumption, the output y of a non-ideal amplifier can be written as a power series of the input signal x:

$$y=...

## Simple Concepts Explained: Fixed-Point

IntroductionMost signal processing intensive applications on FPGA are still implemented relying on integer or fixed-point arithmetic. It is not easy to find the key ideas on quantization, fixed-point and integer arithmetic. In a series of articles, I aim to clarify some concepts and add examples on how things are done in real life. The ideas covered are the result of my professional experience and hands-on projects.

In this article I will present the most fundamental question you...

## Plotting Discrete-Time Signals

A discrete-time sinusoid can have frequency up to just shy of half the sample frequency. But if you try to plot the sinusoid, the result is not always recognizable. For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine. But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph, which does not look like a sine when you connect the dots. We typically want the plot of a...

## Evaluate Window Functions for the Discrete Fourier Transform

The Discrete Fourier Transform (DFT) operates on a finite length time sequence to compute its spectrum. For a continuous signal like a sinewave, you need to capture a segment of the signal in order to perform the DFT. Usually, you also need to apply a window function to the captured signal before taking the DFT [1 - 3]. There are many different window functions and each produces a different approximation of the spectrum. In this post, we’ll present Matlab code that...

## 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.

To 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 magnitude of the cosine wave's...

## Delay estimation by FFT

Given x=sig(t) and y=ref(t), returns [c, ref(t+delta), delta)] = fitSignal(y, x);:Estimates and corrects delay and scaling factor between two signals Code snippetThis article relates to the Matlab / Octave code snippet: Delay estimation with subsample resolution It explains the algorithm and the design decisions behind it.

IntroductionThere are many DSP-related problems, where an unknown timing between two signals needs to be determined and corrected, for example, radar, sonar,...

## Add the Hilbert Transformer to Your DSP Toolkit, Part 1

In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail. In this article, I’ll dig a little deeper into how the Hilbert Transformer works. Understanding the Hilbert Transformer involves a modest amount of mathematics, but the payoff in useful applications is worth it.

As we’ll learn, a Hilbert Transformer is just a particular type of Finite Impulse Response (FIR) filter. In Part 1 of this article, I’ll...

## 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...

## 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...

## Linear-phase DC Removal Filter

This blog describes several DC removal networks that might be of interest to the dsprelated.com readers.

Back in August 2007 there was a thread on the comp.dsp newsgroup concerning the process of removing the DC (zero Hz) component from a time-domain sequence [1]. Discussed in that thread was the notion of removing a signal's DC bias by subtracting the signal's moving average from that signal, as shown in Figure 1(a).

Figure 1.

At first I thought...

## A New Related Site!

We are delighted to announce the launch of the very first new Related site in 15 years! The new site will be dedicated to the trendy and quickly growing field of Machine Learning and will be called - drum roll please - MLRelated.com.

We think MLRelated fits perfectly well within the “Related” family, with:

- the fast growth of TinyML, which is a topic of great interest to the EmbeddedRelated community
- the use of Machine/Deep Learning in Signal Processing applications, which is of...

## Generating pink noise

In one of his most famous columns for Scientific American, Martin Gardner wrote about pink noise and its relation to fractal music. The article was based on a 1978 paper by Voss and Clarke, which presents, among other things, a simple algorithm for generating pink noise, also known as 1/f noise.

The fundamental idea of the algorithm is to add up several sequences of uniform random numbers that get updated at different rates. The first source gets updated at...

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

## Frequency Dependence in Free Space Propagation

Introduction

It seems to be fairly common knowledge, even among practicing professionals, that the efficiency of propagation of wireless signals is frequency dependent. Generally it is believed that lower frequencies are desirable since pathloss effects will be less than they would be at higher frequencies. As evidence of this, the Friis Transmission Equation[i] is often cited, the general form of which is usually written as:

Pr = Pt Gt Gr ( λ / 4πd )2 (1)

where the...

## Embedded Toolbox: Programmer's Calculator

Like any craftsman, I have accumulated quite a few tools during my embedded software development career. Some of them proved to me more useful than others. And these generally useful tools ended up in my Embedded Toolbox. In this blog, I'd like to share some of my tools with you. Today, I'd like to start with my cross-platform Programmer's Calculator called QCalc.

I'm sure that you already have your favorite calculator online or on your smartphone. But can your calculator accept...

## Interpolation Basics

This article covers interpolation basics, and provides a numerical example of interpolation of a time signal. Figure 1 illustrates what we mean by interpolation. The top plot shows a continuous time signal, and the middle plot shows a sampled version with sample time Ts. The goal of interpolation is to increase the sample rate such that the new (interpolated) sample values are close to the values of the continuous signal at the sample times [1]. For example, if...

## Linear-phase DC Removal Filter

This blog describes several DC removal networks that might be of interest to the dsprelated.com readers.

Back in August 2007 there was a thread on the comp.dsp newsgroup concerning the process of removing the DC (zero Hz) component from a time-domain sequence [1]. Discussed in that thread was the notion of removing a signal's DC bias by subtracting the signal's moving average from that signal, as shown in Figure 1(a).

Figure 1.

At first I thought...

## An s-Plane to z-Plane Mapping Example

While surfing around the Internet recently I encountered the 's-plane to z-plane mapping' diagram shown in Figure 1. At first I thought the diagram was neat because it's a good example of the old English idiom: "A picture is worth a thousand words." However, as I continued to look at Figure 1 I began to detect what I believe are errors in the diagram.

Reader, please take a few moments to see if you detect any errors in Figure 1.

...## 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...

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

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

## A Differentiator With a Difference

Some time ago I was studying various digital differentiating networks, i.e., networks that approximate the process of taking the derivative of a discrete time-domain sequence. By "studying" I mean that I was experimenting with various differentiating filter coefficients, and I discovered a computationally-efficient digital differentiator. A differentiator that, for low fequency signals, has the power of George Foreman's right hand! Before I describe this differentiator, let's review a few...

## 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...

## 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 titled "

## Code Snippets Section Now LIVE

The new code sharing section is now live and can be accessed HERE.

Please take a few minutes to rate and/or comment the snippets that you have the expertise to judge.

If you think of some code snippets that you would like to share with the DSP community, please apply to become a contributor HERE.

If you are not aware of the reward program for contributors, your can learn about it HERE.

As always, your comments and suggestions are...

## New Code Sharing Section & Reward Program for Contributors!

UPDATE (11/02/2010): The code section is now live.

UPDATE 2 (01/31/2011): The reward program has changed. A flat fee of $20 per code snippet submitted will now be paid.

_______________

I am very happy to finally announce the imminent launch of the new code sharing section. My vision for this new section is a rich library of high quality code snippets for the DSP community, from processor specific functions to Matlab or Scilab routines, from the simplest filter...

## 50,000th Member Announced!

In my last post, I wrote that DSPRelated.com was about to reach the 50,000 members mark. Well, I am very happy to announce that it happened during the holidays, and the lucky person is Charlie Tsai from Taiwan. Charlie is an assistant professor in the Department of Electrical Engineering at the National Central University in Taiwan where he teaches the "Biomedical Signal Processing" class. He is also the advisor of the

## Almost 50,000 Members!

I am very happy to announce that DSPRelated.com will reach the 50,000 registered members mark before the end of 2009. To celebrate this milestone, I will buy a BMW 5 to the 50,000th person to register (please make sure to confirm you email address to activate your registration). Please read the fine prints after the picture.

I am just having fun here and it's not even April's fool day. The 50,000th member won't get a BMW (I wish I could offer it!),...

## DSPRelated faster than ever!

if you are visiting DSPRelated.com on a regular basis, you should observe that the site loads significantly faster in your browser than it used to, especially if you are in Europe or in Asia. The main reason for this is that I am now using Amazon's CloudFront service for the delivery of most static content on DSPRelated.com (images, javascripts, css). The cloudFront service automatically detects the location of a visitor and will deliver the static content from the server...

## New Papers / Theses Section

The new 'Papers & Theses' section is now online: http://www.dsprelated.com/documents.phpThe idea is to list and organize in one place as many DSP related dissertations (PhD & Masters) and papers/articles as possible.If you are the author of a thesis or paper and would like to have it listed on DSPRelated.com, please follow these steps:- Make sure that you are allowed to share the document online (copyright).- If you don't already have one, make a 'pdf' copy of your document. ...

## New Blog Section!

By now, chances are you have noticed the new blogs section (you are actually in it right now!).

Following an email I sent to the members of the site, a few weeks ago, asking for dsp engineers willing to blog here, I received around 50 propositions. I have selected an initial set of 10 bloggers (that I will soon introduce into a seperate post) and I am currently in the process of creating their accounts. Markus and Parth have already...

## New Discussion Group: DSP & FPGA

I have just created a new discussion group for engineers implementing DSP functions on FPGAs. The creation of this group has been on my todo list for a long time. If you want to join the group, send a blank email to: fpgadsp-subscribe@yahoogroups.com

As usual, it should take a few weeks before there are enough members for interesting discussions to get started.