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...
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...
A Two Bin Solution
Introduction
This is an article to hopefully give a better understanding of the Discrete Fourier Transform (DFT) by showing an implementation of how the parameters of a real pure tone can be calculated from just two DFT bin values. The equations from previous articles are used in tandem to first calculate the frequency, and then calculate the amplitude and phase of the tone. The approach works best when the tone is between the two DFT bins in terms of frequency.
The Coding...Reduced-Delay IIR Filters
This blog gives the results of a preliminary investigation of reduced-delay (reduced group delay) IIR filters based on my understanding of the concepts presented in a recent interesting blog by Steve Maslen [1].
Development of a Reduced-Delay 2nd-Order IIR Filter
Maslen's development of a reduced-delay 2nd-order IIR filter begins with a traditional prototype filter, HTrad, shown in Figure 1(a). The first modification to the prototype filter is to extract the b0 feedforward coefficient...
Part 11. Using -ve Latency DSP to Cancel Unwanted Delays in Sampled-Data Filters/Controllers
This final article in the series will look at -ve latency DSP and how it can be used to cancel the unwanted delays in sampled-data systems due to such factors as Nyquist filtering, ADC acquisition, DSP/FPGA algorithm computation time, DAC reconstruction and circuit propagation delays.
Some applications demand zero-latency or zero unwanted latency signal processing. Negative latency DSP may sound like the stuff of science fiction or broken physics but the arrangement as...
A Direct Digital Synthesizer with Arbitrary Modulus
Suppose you have a system with a 10 MHz sample clock, and you want to generate a sampled sinewave at any frequency below 5 MHz on 500 kHz spacing; i.e., 0.5, 1.0, 1.5, … MHz. In other words, f = k*fs/20, where k is an integer and fs is sample frequency. This article shows how to do this using a simple Direct Digital Synthesizer (DDS) with a look-up table that is at most 20 entries long. We’ll also demonstrate a Quadrature-output DDS. A note on...
Somewhat Off Topic: Deciphering Transistor Terminology
I recently learned something mildly interesting about transistors, so I thought I'd share my new knowledge with you folks. Figure 1 shows a p-n-p transistor comprising a small block of n-type semiconductor sandwiched between two blocks of p-type semiconductor.
The terminology of "emitter" and "collector" seems appropriate, but did you ever wonder why the semiconductor block in the center is called the "base"? The word base seems inappropriate because the definition of the word base is:...
Reducing IIR Filter Computational Workload
This blog describes a straightforward method to significantly reduce the number of necessary multiplies per input sample of traditional IIR lowpass and highpass digital filters.
Reducing IIR Filter Computations Using Dual-Path Allpass Filters
We can improve the computational speed of a lowpass or highpass IIR filter by converting that filter into a dual-path filter consisting of allpass filters as shown in Figure 1.
...A Lesson In Engineering Humility
Let's assume you were given the task to design and build the 12-channel telephone transmission system shown in Figure 1.
Figure 1
At a rate of 8000 samples/second, each telephone's audio signal is sampled and converted to a 7-bit binary sequence of pulses. The analog signals at Figure 1's nodes A, B, and C are presented in Figure 2.
Figure 2
I'm convinced that some of you subscribers to this dsprelated.com web site could accomplish such a design & build task....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...
Data Types for Control & DSP
There's a lot of information out there on what data types to use for digital signal processing, but there's also a lot of confusion, so the topic bears repeating.
I recently posted an entry on PID control. In that article I glossed over the data types used by showing "double" in all of my example code. Numerically, this should work for most control problems, but it can be an extravagant use of processor resources. There ought to be a better way to determine what precision you need...
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...
Fitting a Damped Sine Wave
A damped sine wave is described by
$$ x_{(k)} = A \cdot e^{\alpha \cdot k} \cdot cos(\omega \cdot k + p)\tag{1}$$
with frequency $\omega$ , phase p , initial amplitude A and damping constant $\alpha$ . The $x_{(k)}$ are the samples of the function at equally spaced points in time.
With $x_{(k)}$ given, one often has to find the unknown parameters of the function. This can be achieved for instance with nonlinear approximation or with DFT – methods.
I present a method to find the...
Python scipy.signal IIR Filter Design
Introduction
The following is an introduction on how to design an infinite impulse response (IIR) filters using the Python scipy.signal package. This post, mainly, covers how to use the scipy.signal package and is not a thorough introduction to IIR filter design. For complete coverage of IIR filter design and structure see one of the references.
Filter SpecificationBefore providing some examples lets review the specifications for a filter design. A filter...
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...
Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images
This blog explains why, in the process of time-domain interpolation (sample rate increase), zero stuffing a time sequence with zero-valued samples produces an increased-length time sequence whose spectrum contains replications of the original time sequence's spectrum.
Background
The traditional way to interpolate (sample rate increase) an x(n) time domain sequence is shown in Figure 1.
Figure 1
The '↑ L' operation in Figure 1 means to...
ADC Clock Jitter Model, Part 1 – Deterministic Jitter
Analog to digital converters (ADC’s) have several imperfections that affect communications signals, including thermal noise, differential nonlinearity, and sample clock jitter [1, 2]. As shown in Figure 1, the ADC has a sample/hold function that is clocked by a sample clock. Jitter on the sample clock causes the sampling instants to vary from the ideal sample time. This transfers the jitter from the sample clock to the input signal.
In this article, I present a Matlab...
Feedback Controllers - Making Hardware with Firmware. Part 5. Some FPGA Aspects.
This part of the on-going series of articles looks at a variety of aspects concerning the FPGA device which provides the high-speed maths capability for the low-latency controller and the arbitrary circuit generator application. In due course a complete specification along with application examples will be maintained on the project website here.- Part 5: Some FPGA Aspects (this part)
- Part 4: Engineering of...
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.
...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...
Signal Processing Contest in Python (PREVIEW): The Worst Encoder in the World
When I posted an article on estimating velocity from a position encoder, I got a number of responses. A few of them were of the form "Well, it's an interesting article, but at slow speeds why can't you just take the time between the encoder edges, and then...." My point was that there are lots of people out there which take this approach, and don't take into account that the time between encoder edges varies due to manufacturing errors in the encoder. For some reason this is a hard concept...
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...
Sinusoidal Frequency Estimation Based on Time-Domain Samples
The topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on dsprelated.com. For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact"...
Signed serial-/parallel multiplication
Keywords: Binary signed multiplication implementation, RTL, Verilog, algorithm
Summary- A detailed discussion of bit-level trickstery in signed-signed multiplication
- Algorithm based on Wikipedia example
- Includes a Verilog implementation with parametrized bit width
A straightforward method to multiply two binary numbers is to repeatedly shift the first argument a, and add to a register if the corresponding bit in the other argument b is set. The...
The History of CIC Filters: The Untold Story
If you have ever studied or designed a cascaded integrator-comb (CIC) lowpass filter then surely you've read Eugene Hogenauer's seminal 1981 IEEE paper where he first introduced the CIC filter to the signal processing world [1]. As it turns out, Hogenauer's famous paper was not the first formal document describing and proposing CIC filters. Here's the story.
In the Fall of 1979 Eugene Hogenauer was finalizing his development of the CIC filter, the filter now used in so many multirate signal...
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...
Goertzel Algorithm for a Non-integer Frequency Index
If you've read about the Goertzel algorithm, you know it's typically presented as an efficient way to compute an individual kth bin result of an N-point discrete Fourier transform (DFT). The integer-valued frequency index k is in the range of zero to N-1 and the standard block diagram for the Goertzel algorithm is shown in Figure 1. For example, if you want to efficiently compute just the 17th DFT bin result (output sample X17) of a 64-point DFT you set integer frequency index k = 17 and N =...
Wavelets II - Vanishing Moments and Spectral Factorization
In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.
Besides the content of the last post, you should be familiar with basic complex algebra, the...
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...
Setting the 3-dB Cutoff Frequency of an Exponential Averager
This blog discusses two ways to determine an exponential averager's weighting factor so that the averager has a given 3-dB cutoff frequency. Here we assume the reader is familiar with exponential averaging lowpass filters, also called a "leaky integrators", to reduce noise fluctuations that contaminate constant-amplitude signal measurements. Exponential averagers are useful because they allow us to implement lowpass filtering at a low computational workload per output sample.
Figure 1 shows...
