Generating Partially Correlated Random Variables
IntroductionIt is often useful to be able to generate two or more signals with specific cross-correlations. Or, more generally, we would like to specify an $\left(N \times N\right)$ covariance matrix, $\mathbf{R}_{xx}$, and generate $N$ signals which will produce this covariance matrix.
There are many applications in which this technique is useful. I discovered a version of this method while analysing radar systems, but the same approach can be used in a very wide range of...
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...
Angle Addition Formulas from Euler's Formula
Introduction
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT), but only indirectly. The main intent is to get someone who is uncomfortable with complex numbers a little more used to them and relate them back to already known Trigonometric relationships done in Real values. It is essentially a followup to my first blog article "The Exponential Nature of the Complex Unit Circle".
Polar CoordinatesThe more common way of...
Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT
One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate. The derivation of can be found in textbooks [1,2]. You can also demonstrate this principle numerically using the Discrete Fourier Transform (DFT).
The DFT of the sampled signal x(n) is defined as:
$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$
Where
X(k) = discrete frequency spectrum of time sequence x(n)
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...
Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'
In a recent Forum post here on dsprelated.com the audio signal processing subject of stereophonic amplitude-panning was discussed. And in that Forum thread the so-called "Tangent Law", the fundamental principle of stereophonic amplitude-panning, was discussed. However, none of the Forum thread participants had ever seen a derivation of the Tangent Law. This blog presents such a derivation and if this topic interests you, then please read on.
The notion of stereophonic amplitude-panning is...
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...
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...
Beat Notes: An Interesting Observation
Some weeks ago a friend of mine, a long time radio engineer as well as a piano player, called and asked me,
"When I travel in a DC-9 aircraft, and I sit back near the engines, I hear this fairly loud unpleasant whump whump whump whump sound. The frequency of that sound is, maybe, two cycles per second. I think that sound is a beat frequency because the DC-9's engines are turning at a slightly different number of revolutions per second. My question is, what sort of mechanism in the airplane...
Angle Addition Formulas from Euler's Formula
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT), but only indirectly. The main intent is to get someone who is uncomfortable with complex numbers a little more used to them and relate them back to already known Trigonometric relationships done in Real values. It is essentially a followup to my first blog article "The Exponential Nature of the Complex Unit Circle".
Polar CoordinatesThe more common way 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...
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...
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 Processing and
IIR Bandpass Filters Using Cascaded Biquads
In an earlier post [1], we implemented lowpass IIR filters using a cascade of second-order IIR filters, or biquads.
This post provides a Matlab function to do the same for Butterworth bandpass IIR filters. Compared to conventional implementations, bandpass filters based on biquads are less sensitive to coefficient quantization [2]. This becomes important when designing narrowband filters.
A biquad section block diagram using the Direct Form II structure [3,4] is...
Phase and Amplitude Calculation for a Pure Real Tone in a DFT: Method 1
IntroductionThis is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by deriving exact formulas for the phase and amplitude of a non-integer frequency real tone in a DFT. The linearity of the Fourier Transform is exploited to reframe the problem as the equivalent of finding a set of coordinates in a specific vector space. The found coordinates are then used to calculate the phase and amplitude of the pure real tone in the DFT. This article...
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...
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...
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...
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...
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...
Computing Large DFTs Using Small FFTs
It is possible to compute N-point discrete Fourier transforms (DFTs) using radix-2 fast Fourier transforms (FFTs) whose sizes are less than N. For example, let's say the largest size FFT software routine you have available is a 1024-point FFT. With the following trick you can combine the results of multiple 1024-point FFTs to compute DFTs whose sizes are greater than 1024.
The simplest form of this idea is computing an N-point DFT using two N/2-point FFT operations. Here's how the trick...
How Discrete Signal Interpolation Improves D/A Conversion
This blog post is also available in pdf format. Download here.Earlier this year, for the Linear Audio magazine, published in the Netherlands whose subscribers are technically-skilled hi-fi audio enthusiasts, I wrote an article on the fundamentals of interpolation as it's used to improve the performance of analog-to-digital conversion. Perhaps that article will be of some value to the subscribers of dsprelated.com. Here's what I wrote:
We encounter the process of digital-to-analog...
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:...
TCP/IP interface (Matlab/Octave)
Communicate with measurement instruments via Ethernet (no-toolbox-Matlab or Octave)
PurposeMeasurement automation is digital signal processing in a wider sense: Getting a digital signal from an analog world usually involves some measurement instruments, for example a spectrum analyzer. Modern instruments, and also many off-the-shelf prototyping boards such as FPGA cards [1] or microcontrollers [2] are able to communicate via Ethernet. Here, I provide some basic mex-functions (compiled C...
Phase or Frequency Shifter Using a Hilbert Transformer
In this article, we’ll describe how to use a Hilbert transformer to make a phase shifter or frequency shifter. In either case, the input is a real signal and the output is a real signal. We’ll use some simple Matlab code to simulate these systems. After that, we’ll go into a little more detail on Hilbert transformer theory and design.
Phase ShifterA conceptual diagram of a phase shifter is shown in Figure 1, where the bold lines indicate complex...
Spectral Flipping Around Signal Center Frequency
Most of us are familiar with the process of flipping the spectrum (spectral inversion) of a real signal by multiplying that signal's time samples by (-1)n. In that process the center of spectral rotation is fs/4, where fs is the signal's sample rate in Hz. In this blog we discuss a different kind of spectral flipping process.
Consider the situation where we need to flip the X(f) spectrum in Figure 1(a) to obtain the desired Y(f) spectrum shown in Figure 1(b). Notice that the center of...
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 Verilog-A in their...
