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=...
A Narrow Bandpass Filter in Octave or Matlab
The design of a very narrow bandpass FIR filter, coded in either Octave or Matlab, can prove challenging if a computationally-efficient filter is required. This is especially true if the sampling rate is high relative to the filter's center frequency. The most obvious filter design methods, using either window-based or Remez ( Parks-McClellan ) functions, can easily result in filters with many thousands of taps.
The filter to be described reduces the computational effort (and thus...
Second Order Discrete-Time System Demonstration
Discrete-time systems are remarkable: the time response can be computed from mere difference equations, and the coefficients ai, bi of these equations are also the coefficients of H(z). Here, I try to illustrate this remarkableness by converting a continuous-time second-order system to an approximately equivalent discrete-time system. With a discrete-time model, we can then easily compute the time response to any input. But note that the goal here is as much to...
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...
Are DSPs Dead ?
Are DSPs Dead ?Former Texas Instruments Sr. Fellow Gene Frantz and former TI Fellow Alan Gatherer wrote a 2017 IEEE article about the "death and rebirth" of DSP as a discipline, explaining that now signal processing provides indispensable building blocks in widely popular and lucrative areas such as data science and machine learning. The article implies that DSP will now be taught in university engineering programs as its linear systems and electromagnetics...
Digging into an Audio Signal and the DSP Process Pipeline
In this post, I'll look at the benefits of using multiple perspectives when handling signals.A Pre-existing Audio File
Let's say we have an audio file of interest. Let's load it into Audacity and zoom in a little (using View → Zoom → Zoom In, multiple times). The figure illustrates the audio signal: just a basic single-tone signal.
By continuing to zoom into the signal, we eventually get to the point of seeing individual samples as illustrated below. Notice that I've marked one...
A Simplified Matlab Function for Power Spectral Density
In an earlier post [1], I showed how to compute power spectral density (PSD) of a discrete-time signal using the Matlab function pwelch [2]. Pwelch is a useful function because it gives the correct output, and it has the option to average multiple Discrete Fourier Transforms (DFTs). However, a typical function call has five arguments, and it can be hard to remember how to set them all and how they default.
In this post, I create a simplified PSD function by putting a...
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...
Learn to Use the Discrete Fourier Transform
Discrete-time sequences arise in many ways: a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter. We may wish to know the frequency spectrum of any of these sequences. The most-used tool to accomplish this is the Discrete Fourier Transform (DFT), which computes the discrete frequency spectrum of a discrete-time sequence. The DFT is easily calculated using software, but applying it successfully can be challenging. This article provides Matlab examples of some techniques you can use to obtain useful DFT’s.
Sampling bandpass signals
Sampling bandpass signals 1.1 Introduction
It 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...
FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending
This blog presents several interesting things I recently learned regarding the estimation of a spectral value located at a frequency lying between previously computed FFT spectral samples. My curiosity about this FFT interpolation process was triggered by reading a spectrum analysis paper written by three astronomers [1].
My fixation on one equation in that paper led to the creation of this blog.
Background
The notion of FFT interpolation is straightforward to describe. That is, for example,...
Design a DAC sinx/x Corrector
This post provides a Matlab function that designs linear-phase FIR sinx/x correctors. It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.
A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process. In DSP math, we treat the digital signal applied to the DAC is a sequence of impulses. These are converted by the DAC into contiguous pulses...
The 2024 DSP Online Conference
We are very excited to announce that the DSP Online Conference is back this year for a fourth year in a row and will take place October 29, 30 and 31.
Unlike traditional DSP conferences, where most talks are highly specialized and tailored to researchers, our conference is designed to be accessible to a broader audience of DSP enthusiasts, from students and practicing engineers to hobbyists and DSP experts.
For this year's edition, we are aiming to provide a program that will be organized...
The Power Spectrum
Often, when calculating the spectrum of a sampled signal, we are interested in relative powers, and we don’t care about the absolute accuracy of the y axis. However, when the sampled signal represents an analog signal, we sometimes need an accurate picture of the analog signal’s power in the frequency domain. This post shows how to calculate an accurate power spectrum.
Parseval’s theorem [1,2] is a property of the Discrete Fourier Transform (DFT) that...
DFT Bin Value Formulas for Pure Real Tones
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:
SAVE THE DATE – DSPRelated’s First Ever In-Person Conference!
After 25 years of running DSPRelated and more recently, co-organizing the DSP Online Conference, I've never been more excited to share an announcement: we're going in-person!
That's right! We strongly believe this DSP community is ripe for a face-to-face gathering. Our very first in-person conference will be held in Silicon Valley from October 14-16, 2025.
This won’t be just another conference. It will be:
- Organized by engineers, for engineers
- Designed to be empowering,...
Polyphase filter / Farrows interpolation
Hello,
this article is meant to give a quick overview over polyphase filtering and Farrows interpolation.
A good reference with more depth is for example Fred Harris' paper: http://www.signumconcepts.com/IP_center/paper018.pdf
The task is as follows: Interpolate a band-limited discrete-time signal at a variable offset between samples.In other words:Delay the signal by a given amount with sub-sample accuracy.Both mean the same.
The picture below shows samples (black) representing...
The Discrete Fourier Transform and the Need for Window Functions
The Discrete Fourier Transform (DFT) is used to find the frequency spectrum of a discrete-time signal. A computationally efficient version called the Fast Fourier Transform (FFT) is normally used to calculate the DFT. But, as many have found to their dismay, the FFT, when used alone, usually does not provide an accurate spectrum. The reason is a phenomenon called spectral leakage.
Spectral leakage can be reduced drastically by using a window function in conjunction...
Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction
Last time, we talked about error correction and detection, covering some basics like Hamming distance, CRCs, and Hamming codes. If you are new to this topic, I would strongly suggest going back to read that article before this one.
This time we are going to cover Reed-Solomon codes. (I had meant to cover this topic in Part XV, but the article was getting to be too long, so I’ve split it roughly in half.) These are one of the workhorses of error-correction, and they are used in...
Computing the Group Delay of a Filter
I just learned a new method (new to me at least) for computing the group delay of digital filters. In the event this process turns out to be interesting to my readers, this blog describes the method. Let's start with a bit of algebra so that you'll know I'm not making all of this up.
Assume we have the N-sample h(n) impulse response of a digital filter, with n being our time-domain index, and that we represent the filter's discrete-time Fourier transform (DTFT), H(ω), in polar form...
Digital PLL's -- Part 2
In Part 1, we found the time response of a 2nd order PLL with a proportional + integral (lead-lag) loop filter. Now let’s look at this PLL in the Z-domain [1, 2]. We will find that the response is characterized by a loop natural frequency ωn and damping coefficient ζ.
Having a Z-domain model of the DPLL will allow us to do three things:
Compute the values of loop filter proportional gain KL and integrator gain KI that give the desired loop natural...Take Control of Noise with Spectral Averaging
Most engineers have seen the moment-to-moment fluctuations that are common with instantaneous measurements of a supposedly steady spectrum. You can see these fluctuations in magnitude and phase for each frequency bin of your spectrogram. Although major variations are certainly reason for concern, recall that we don’t live in an ideal, noise-free world. After verifying the integrity of your measurement setup by checking connections, sensors, wiring, and the like, you might conclude that the...
Noise shaping
eywords: Quantization noise; noise shaping
A brief introduction to noise shaping, with firm resolve not to miss the forest for the trees. We may still stumble over some assorted roots. Matlab example code is included.
QuantizationFig. 1 shows a digital signal that is reduced to a lower bit width, for example a 16 bit signal being sent to a 12 bit digital-to-analog converter. Rounding to the nearest output value is obviously the best that can be done to minimize the error of each...
Accurate Measurement of a Sinusoid's Peak Amplitude Based on FFT Data
There are two code snippets associated with this blog post:
and
Testing the Flat-Top Windowing Function
This blog discusses an accurate method of estimating time-domain sinewave peak amplitudes based on fast Fourier transform (FFT) data. Such an operation sounds simple, but the scalloping loss characteristic of FFTs complicates the process. We eliminate that complication by...
Fractional Delay FIR Filters
Consider the following Finite Impulse Response (FIR) coefficients:
b = [b0 b1 b2 b1 b0]
These coefficients form a 5-tap symmetrical FIR filter having constant group delay [1,2] over 0 to fs/2 of:
D = (ntaps – 1)/2 = 2 samples
For a symmetrical filter with an odd number of taps, the group delay is always an integer number of samples, while for one with an even number of taps, the group delay is always an integer + 0.5 samples. Can we design a filter...
Oscilloscope Dreams
My coworkers and I recently needed a new oscilloscope. I thought I would share some of the features I look for when purchasing one.
When I was in college in the early 1990's, our oscilloscopes looked like this:
Now the cathode ray tubes have almost all been replaced by digital storage scopes with color LCD screens, and they look like these:
Oscilloscopes are basically just fancy expensive boxes for graphing voltage vs. time. They span a wide range of features and prices:...
Padé Delay is Okay Today
This article is going to be somewhat different in that I’m not really writing it for the typical embedded systems engineer. Rather it’s kind of a specialized topic, so don’t be surprised if you get bored and move on to something else. That’s fine by me.
Anyway, let’s just jump ahead to the punchline. Here’s a numerical simulation of a step response to a \( p=126, q=130 \) Padé approximation of a time delay:
Impressed? Maybe you should be. This...
An Efficient Linear Interpolation Scheme
This blog presents a computationally-efficient linear interpolation trick that requires at most one multiply per output sample.
Background: Linear Interpolation
Looking at Figure 1(a) let's assume we have two points, [x(0),y(0)] and [x(1),y(1)], and we want to compute the value y, on the line joining those two points, associated with the value x.
Figure 1: Linear interpolation: given x, x(0), x(1), y(0), and y(1), compute the value of y. ...
