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 FileLet'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...
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...
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...
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. ...
Summary of ROC Rules
This is a very short guide on how to find all possible outcomes of a system where Region of Convergence (ROC) and the original signal is not known.
Access to 50+ Sessions From the DSP Online Conference
In case you forget or didn't already know, registering for the 2023 DSP Online Conference automatically gives you 10 months of unlimited access to all sessions from previous editions of the conference. So for the price of an engineering book, you not only get access to the upcoming 2023 DSP Online Conference but also to hours upon hours of on-demand DSP gold from some of the best experts in the field.
The value you get for your small investment is simply huge. Many of the...
Compute the Frequency Response of a Multistage Decimator
Figure 1a shows the block diagram of a decimation-by-8 filter, consisting of a low-pass finite impulse response (FIR) filter followed by downsampling by 8 [1]. A more efficient version is shown in Figure 1b, which uses three cascaded decimate-by-two filters. This implementation has the advantages that only FIR 1 is sampled at the highest sample rate, and the total number of filter taps is lower.
The frequency response of the single-stage decimator before downsampling is just...
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,...
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
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...
5G NR QC-LDPC Encoding Algorithm
3GPP 5G has been focused on structured LDPC codes known as quasi-cyclic low-density parity-check (QC-LDPC) codes, which exhibit advantages over other types of LDPC codes with respect to the hardware implementations of encoding and decoding using simple shift registers and logic circuits.
5G NR QC-LDPC Circulant Permutation MatrixA circular permutation matrix ${\bf I}(P_{i,j})$ of size $Z_c \times Z_c$ is obtained by circularly shifting the identity matrix $\bf I$ of...
An Astounding Digital Filter Design Application
I've recently encountered a digital filter design application that astonished me with its design flexibility, capability, and ease of use. The software is called the "ASN Filter Designer." After experimenting with a demo version of this filter design software I was so impressed that I simply had publicize it to the subscribers here on dsprelated.com.
What I Liked About the ASN Filter DesignerWith typical filter design software packages the user enters numerical values for the...
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...
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...
Frequency-Domain Periodicity and the Discrete Fourier Transform
Introduction
Some of the better understood aspects of time-sampled systems are the limitations and requirements imposed by the Nyquist sampling theorem [1]. Somewhat less understood is the periodic nature of the spectra of sampled signals. This article provides some insights into sampling that not only explain the periodic nature of the sampled spectrum, but aliasing, bandlimited sampling, and the so-called "super-Nyquist" or IF sampling. The approaches taken here include both mathematical...
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...
Time Machine, Anyone?
Abstract: Dispersive linear systems with negative group delay have caused much confusion in the past. Some claim that they violate causality, others that they are the cause of superluminal tunneling. Can we really receive messages before they are sent? This article aims at pouring oil in the fire and causing yet more confusion :-).
IntroductionIn this article we reproduce the results of a physical experiment...
The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase
This blog discusses a little-known filter characteristic that enables real- and complex-coefficient tapped-delay line FIR filters to exhibit linear phase behavior. That is, this blog answers the question:
What is the constraint on real- and complex-valued FIR filters that guarantee linear phase behavior in the frequency domain?I'll declare two things to convince you to continue reading.
Declaration# 1: "That the coefficients must be symmetrical" is not a correct
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...
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...
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...
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...