## Sampling bandpass signals

Sampling bandpass signals 1.1 IntroductionIt is known [1], [3] that bandpass signals can be sampled with a sampling frequency which is lower than the sampling frequency according to the sampling theorem.

Fig. 1 shows an example of how the spectrum of a bandpass signal sampled with $f_s$ (Fig. 1a) arises in the baseband with $−f_s / 2 ≤ f < f_s/2$. The bandpass signal is assumed to have a center frequency $f_c = (f_{max} + f_{min})/2$ and bandwidth $\Delta f...

## Digital Filter Instructions from IKEA?

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and play.

Swedish “Bygglek” = build and play. Swedish “Bygglek” = build and...

## Simulink-Simulation of SSB demodulation

≥≥≥ Simulink-Simulation of SSB demodulation or modulation from the article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons Josef HoffmannThe article “Understanding the ‘Phasing Method’ of Single Sideband Demodulation” by Richard Lyons is a very good description of this topic. The block representation from the figures are clear and easy to understand. They are predestined for a simulation in Simulink. The simulation can help...

## Setting Carrier to Noise Ratio in Simulations

When simulating digital receivers, we often want to check performance with added Gaussian noise. In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N). I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.

Definition of C/NThe Carrier to noise ratio is defined as the ratio of average signal power to noise power for a modulated...

## An Efficient Full-Band Sliding DFT Spectrum Analyzer

In this blog I present two computationally efficient full-band discrete Fourier transform (DFT) networks that compute the 0th bin and all the positive-frequency bin outputs for an N-point DFT in real-time on a sample-by-sample basis.

An Even-N Spectrum Analyzer

The full-band sliding DFT (SDFT) spectrum analyzer network, where the DFT size N is an even integer, is shown in Figure 1(a). The x[n] input sequence is restricted to be real-only valued samples. Notice that the only real parts of...

## Update to a Narrow Bandpass Filter in Octave or Matlab

Following my earlier blog post (June 2020) featuring a Narrow Bandpass Filter, I’ve had some useful feedback and suggestions. This has inspired me to come up with an updated version, incorporating the following changes compared to the earlier one :

- Simpler code in Octave or Matlab
- Float32 precision replaces float64
- Faster processing by a factor of at least 4 times
- Easier setup of input parameters
- Normalized signal output level

A new experimental version in...

## Add a Power Marker to a Power Spectral Density (PSD) Plot

Perhaps we should call most Power Spectral Density (PSD) calculations relative PSD, because usually we don’t have to worry about absolute power levels. However, for cases (e.g., measurements or simulations) where we are concerned with absolute power, it would be nice to be able to display it on a PSD plot. Unfortunately, you can’t read the power directly from the plot. For example, the plotted spectral peak of a narrowband signal, such as a sinewave, is lower than the...

## A Simpler Goertzel Algorithm

In this blog I propose a Goertzel algorithm that is simpler than the version of the Goertzel algorithm that is traditionally presented DSP textbooks. Below I very briefly describe the DSP textbook version of the Goertzel algorithm followed by a description of my proposed simpler algorithm.

The Traditional DSP Textbook Goertzel Algorithm

The so-called Goertzel algorithm is used to efficiently compute a single mth-bin sample of an N-point discrete Fourier transform (DFT) [1-4]. The...

## 60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing

Electrocardiogram (ECG) signals are obtained by monitoring the electrical activity of the human heart for medical diagnostic purposes [1]. This blog describes a very efficient digital filter used to reduce both 60 Hz AC power line noise and unwanted signal baseline drift that often contaminate ECG signals.

PDF_HERE

We'll first describe the ECG noise reduction filter and then examine the filter's performance in a real-world ECG signal filtering example.Proposed ECG Noise Reduction Digital...

## 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...

## 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...

## Understanding and Preventing Overflow (I Had Too Much to Add Last Night)

Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.

In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:

for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...## A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters

This blog discusses the behavior, mathematics, and implementation of cascaded integrator-comb filters.

Cascaded integrator-comb (CIC) digital filters are computationally-efficient implementations of narrowband lowpass filters, and are often embedded in hardware implementations of decimation, interpolation, and delta-sigma converter filtering.

After describing a few applications of CIC filters, this blog introduces their structure and behavior, presents the frequency-domain...

## 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...

## 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 fractional components. The bit length is defined...

## Computing FFT Twiddle Factors

Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).

Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...

## 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...

## Minimum Shift Keying (MSK) - A Tutorial

Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to non-linear distortion and was therefore chosen as the modulation technique for the GSM cell phone standard.

MSK is a special case of Continuous-Phase Frequency Shift Keying (CPFSK) which is a special case of a general class of modulation schemes known as Continuous-Phase Modulation (CPM). It is worth noting that CPM (and hence CPFSK) is a...

## Design IIR Butterworth Filters Using 12 Lines of Code

While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch. You can do it in 12 lines of Matlab code. In this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order. Here is an example function call for a 5th order filter:

N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...## Use Matlab Function pwelch to Find Power Spectral Density – or Do It Yourself

In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1]. These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies. The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the so-called Welch power spectral density estimate [3,4].

In this article, I’ll present some...

## 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 fractional components. The bit length is defined...

## 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...

## Adventures in Signal Processing with Python

Author’s note: This article was originally called Adventures in Signal Processing with Python (MATLAB? We don’t need no stinkin' MATLAB!) — the allusion to The Treasure of the Sierra Madre has been removed, in deference to being a good neighbor to The MathWorks. While I don’t make it a secret of my dislike of many aspects of MATLAB — which I mention later in this article — I do hope they can improve their software and reduce the price. Please note this...

## Understanding and Preventing Overflow (I Had Too Much to Add Last Night)

Happy Thanksgiving! Maybe the memory of eating too much turkey is fresh in your mind. If so, this would be a good time to talk about overflow.

In the world of floating-point arithmetic, overflow is possible but not particularly common. You can get it when numbers become too large; IEEE double-precision floating-point numbers support a range of just under 21024, and if you go beyond that you have problems:

for k in [10, 100, 1000, 1020, 1023, 1023.9, 1023.9999, 1024]: try: ...## Ten Little Algorithms, Part 2: The Single-Pole Low-Pass Filter

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 3: Welford's Method (And Friends)
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

I’m writing this article in a room with a bunch of other people talking, and while sometimes I wish they would just SHUT UP, it would be...

## 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'...

## 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...

## 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 ↔...

## Digital Envelope Detection: The Good, the Bad, and the Ugly

Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.

Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...

## Design IIR Butterworth Filters Using 12 Lines of Code

While there are plenty of canned functions to design Butterworth IIR filters [1], it’s instructive and not that complicated to design them from scratch. You can do it in 12 lines of Matlab code. In this article, we’ll create a Matlab function butter_synth.m to design lowpass Butterworth filters of any order. Here is an example function call for a 5th order filter:

N= 5 % Filter order fc= 10; % Hz cutoff freq fs= 100; % Hz sample freq [b,a]=...## 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...

## Already 3000+ Attendees Registered for the Upcoming Embedded Online Conference

Chances are you already know, through the newsletter or banners on the Related sites, about the upcoming Embedded Online Conference.

Chances are you also already know that you have until the end of the month of February to register for free.

And chances are that you are one of the more than 3000 pro-active engineers who have already registered.

But If you are like me and have a tendency to do tomorrow what can be done today, maybe you haven't registered yet. You may...

## Free Goodies from Embedded World - Full Inventory and Upcoming Draw Live-Streaming Date

Chances are that you already know that I went to Embedded World a few weeks ago and came back with a bag full of "goodies". Initially, my vision was to do a single draw for one person to win it all, but I didn't expect to come back with so much stuff and so many development kits. Based on your feedback, it seems like you guys agree that It wouldn't make sense for one person to win everything as no-one could make good use of all the boards and there would be lots of...

## Free Goodies from Embedded World - What to Do Next?

I told you I would go on a hunt for free stuff at Embedded World in order to build a bundle for someone to win.

## Back from Embedded World 2019 - Funny Stories and Live-Streaming Woes

When the idea of live-streaming parts of Embedded World came to me, I got so excited that I knew I had to make it happen. I perceived the opportunity as a win-win-win-win.

- win #1 - Engineers who could not make it to Embedded World would be able to sample the huge event,
- win #2 - The organisation behind EW would benefit from the extra exposure
- win #3 - Lecturers and vendors who would be live-streamed would reach a (much) larger audience
- win #4 - I would get...

## Spread the Word and Run a Chance to Win a Bundle of Goodies from Embedded World

Do you have a Twitter and/or Linkedin account?

If you do, please consider paying close attention for the next few days to the EmbeddedRelated Twitter account and to my personal Linkedin account (feel free to connect). This is where I will be posting lots of updates about how the EmbeddedRelated.tv live streaming experience is going at Embedded World.

The most successful this live broadcasting experience will be, the better the chances that I will be able to do it...

## Launch of EmbeddedRelated.tv

With the upcoming Embedded Word just around the corner, I am very excited to launch the EmbeddedRelated.tv platform.

This is where you will find the schedule for all the live broadcasts that I will be doing from Embedded World next week. Please note that the schedule will be evolving constantly, even during the show, so I suggest your refresh the page often. For instance, I am still unsure if I will be able to do the 'opening of the doors' broadcast as...

## Live Streaming from Embedded World!

For those of you who won't be attending Embedded World this year, I will try to be your eyes and ears by video streaming live from the show floor.

I am not talking improvised streaming from a phone, but real, high quality HD streaming with a high-end camera and a device that will bond three internet connections (one wifi and two cellular) to ensure a steady, and hopefully reliable, stream. All this to hopefully give those of you who cannot be there in person a virtual...

## Sensors Expo - Trip Report & My Best Video Yet!

This was my first time at Sensors Expo and my second time in Silicon Valley and I must say I had a great time.

Before I share with you what I find to be, by far, my best 'highlights' video yet for a conference/trade show, let me try to entertain you with a few anecdotes from this trip. If you are not interested by my stories or maybe don't have the extra minutes needed to read them, please feel free to skip to the end of this blog post to watch the...