A New Contender in the Digital Differentiator Race
This blog proposes a novel differentiator worth your consideration. Although simple, the differentiator provides a fairly wide 'frequency range of linear operation' and can be implemented, if need be, without performing numerical multiplications.
BackgroundIn reference [1] I presented a computationally-efficient tapped-delay line digital differentiator whose $h_{ref}(k)$ impulse response is:
$$ h_{ref}(k) = {-1/16}, \ 0, \ 1, \ 0, \ {-1}, \ 0, \ 1/16 \tag{1} $$and...
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
Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm
If you need to compute inverse fast Fourier transforms (inverse FFTs) but you only have forward FFT software (or forward FFT FPGA cores) available to you, below are four ways to solve your problem.
Preliminaries To define what we're thinking about here, an N-point forward FFT and an N-point inverse FFT are described by:
$$ Forward \ FFT \rightarrow X(m) = \sum_{n=0}^{N-1} x(n)e^{-j2\pi nm/N} \tag{1} $$ $$ Inverse \ FFT \rightarrow x(n) = {1 \over N} \sum_{m=0}^{N-1}...Correcting an Important Goertzel Filter Misconception
Recently I was on the Signal Processing Stack Exchange web site (a question and answer site for DSP people) and I read a posted question regarding Goertzel filters [1]. One of the subscribers posted a reply to the question by pointing interested readers to a Wikipedia web page discussing Goertzel filters [2]. I noticed the Wiki web site stated that a Goertzel filter:
"...is marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling
Analog Devices Inc. has posted a neat software simulation tool on their corporate web site that graphically shows the aliasing effects of both lowpass and bandpass periodic sampling. This is a nice tutorial tool for beginners in DSP.
The tool shows four important characteristics of periodic sampling:
Characteristic# 1: All input analog spectral components, regardless of their center frequencies, show up (appear) below half the sample rate in the digitized...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...
Complex Down-Conversion Amplitude Loss
This blog illustrates the signal amplitude loss inherent in a traditional complex down-conversion system. (In the literature of signal processing, complex down-conversion is also called "quadrature demodulation.")
The general idea behind complex down-conversion is shown in Figure 1(a). And the traditional hardware block diagram of a complex down-converter is shown in Figure 1(b).
Let's assume the input to our down-conversion system is an analog radio frequency (RF) signal,...
A Complex Variable Detective Story – A Disconnect Between Theory and Implementation
Recently I was in the middle of a pencil-and-paper analysis of a digital 5-tap FIR filter having complex-valued coefficients and I encountered a surprising and thought-provoking problem. So that you can avoid the algebra difficulty I encountered, please read on.
A Surprising Algebra Puzzle
I wanted to derive the H(ω) equation for the frequency response of my FIR digital filter whose complex coefficients were h0, h1, h2, h3, and h4. I could then test the validity of my H(ω)...
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...
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 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...
A Simple Complex Down-conversion Scheme
Recently I was experimenting with complex down-conversion schemes. That is, generating an analytic (complex) version, centered at zero Hz, of a real bandpass signal that was originally centered at ±fs/4 (one fourth the sample rate). I managed to obtain one such scheme that is computationally efficient, and it might be of some mild interest to you guys. The simple complex down-conversion scheme is shown in Figure 1(a).It works like this: say we have a real xR(n) input bandpass...
Errata for the book: 'Understanding Digital Signal Processing'
Errata 3rd Ed. International Version.pdfErrata 3rd Ed. International Version.pdfThis blog post provides, in one place, the errata for each of the many different Editions/Printings of my book Understanding Digital Signal Processing.
If you would like the errata for your copy of the book, merely scroll down and click on the appropriate red line below. For the American versions of the various Editions of the book you'll need to know the "Printing Number" of your copy of the...
Correcting an Important Goertzel Filter Misconception
Recently I was on the Signal Processing Stack Exchange web site (a question and answer site for DSP people) and I read a posted question regarding Goertzel filters [1]. One of the subscribers posted a reply to the question by pointing interested readers to a Wikipedia web page discussing Goertzel filters [2]. I noticed the Wiki web site stated that a Goertzel filter:
"...is marginally stable and vulnerable tonumerical error accumulation when computed usinglow-precision arithmetic and...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 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...
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...
Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals
In digital signal processing (DSP) we're all familiar with the processes of bandpass sampling an analog bandpass signal and downsampling a digital bandpass signal. The overall spectral behavior of those operations are well-documented. However, mathematical expressions for computing the translated frequency of individual spectral components, after bandpass sampling or downsampling, are not available in the standard DSP textbooks. The following three sections explain how to compute the...
Improved Narrowband Lowpass IIR Filters
Here's a neat IIR filter trick. It's excerpted from the "DSP Tricks" chapter of the new 3rd edition of my book "Understanding Digital Signal Processing". Perhaps this trick will be of some value to the subscribers of dsprelated.com.
Due to their resistance to quantized-coefficient errors, traditional 2nd-order infinite impulse response (IIR) filters are the fundamental building blocks in computationally-efficient high-order IIR digital filter implementations. However, when used in...
Two Easy Ways To Test Multistage CIC Decimation Filters
This blog presents two very easy ways to test the performance of multistage cascaded integrator-comb (CIC) decimation filters [1]. Anyone implementing CIC filters should take note of the following proposed CIC filter test methods.
Introduction
Figure 1 presents a multistage decimate by D CIC filter where the number of stages is S = 3. The '↓D' operation represents downsampling by integer D (discard all but every Dth sample), n is the input time index, and m is the output time index.
The DFT of Finite-Length Time-Reversed Sequences
Recently I've been reading papers on underwater acoustic communications systems and this caused me to investigate the frequency-domain effects of time-reversal of time-domain sequences. I created this blog because there is so little coverage of this topic in the literature of DSP.
This blog reviews the two types of time-reversal of finite-length sequences and summarizes their discrete Fourier transform (DFT) frequency-domain characteristics.The Two Types of Time-Reversal in DSP
...Do Multirate Systems Have Transfer Functions?
The following text describes why I ask the strange question in the title of this blog. Some months ago I was asked to review a article manuscript, for possible publication in a signal processing journal, that presented a method for improving the performance of cascaded integrator-comb (CIC) decimation filters [1].
Thinking about such filters, Figure 1(a) shows the block diagram of a traditional 2nd-order CIC decimation filter followed by downsampling by the sample rate factor R. There we...
Multiplying Two Binary Numbers
I just encountered what I think is an interesting technique for multiplying two integer numbers. Perhaps some of the readers here will also find it interesting.
Here's the technique: assume we want to multiply 18 times 17. We start by writing 18 and 17, side-by-side in column A and column B, as shown at the top of Figure 1. Next we divide the 18 at the top of column A by two, retaining only the integer part of the division, and double the 17 at the top of column B. The results of those two...
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...
Improved Narrowband Lowpass IIR Filters
Here's a neat IIR filter trick. It's excerpted from the "DSP Tricks" chapter of the new 3rd edition of my book "Understanding Digital Signal Processing". Perhaps this trick will be of some value to the subscribers of dsprelated.com.
Due to their resistance to quantized-coefficient errors, traditional 2nd-order infinite impulse response (IIR) filters are the fundamental building blocks in computationally-efficient high-order IIR digital filter implementations. However, when used in...
A New Contender in the Digital Differentiator Race
This blog proposes a novel differentiator worth your consideration. Although simple, the differentiator provides a fairly wide 'frequency range of linear operation' and can be implemented, if need be, without performing numerical multiplications.
BackgroundIn reference [1] I presented a computationally-efficient tapped-delay line digital differentiator whose $h_{ref}(k)$ impulse response is:
$$ h_{ref}(k) = {-1/16}, \ 0, \ 1, \ 0, \ {-1}, \ 0, \ 1/16 \tag{1} $$and...
Two Easy Ways To Test Multistage CIC Decimation Filters
This blog presents two very easy ways to test the performance of multistage cascaded integrator-comb (CIC) decimation filters [1]. Anyone implementing CIC filters should take note of the following proposed CIC filter test methods.
Introduction
Figure 1 presents a multistage decimate by D CIC filter where the number of stages is S = 3. The '↓D' operation represents downsampling by integer D (discard all but every Dth sample), n is the input time index, and m is the output time index.
Errata for the book: 'Understanding Digital Signal Processing'
Errata 3rd Ed. International Version.pdfErrata 3rd Ed. International Version.pdfThis blog post provides, in one place, the errata for each of the many different Editions/Printings of my book Understanding Digital Signal Processing.
If you would like the errata for your copy of the book, merely scroll down and click on the appropriate red line below. For the American versions of the various Editions of the book you'll need to know the "Printing Number" of your copy of the...
How Not to Reduce DFT Leakage
This blog describes a technique to reduce the effects of spectral leakage when using the discrete Fourier transform (DFT).
In late April 2012 there was a thread on the comp.dsp newsgroup discussing ways to reduce the spectral leakage problem encountered when using the DFT. One post in that thread caught my eye [1]. That post referred to a website presenting a paper describing a DFT leakage method that I'd never heard of before [2]. (Of course, not that I've heard...
Multiplierless Exponential Averaging
This blog discusses an interesting approach to exponential averaging. To begin my story, a traditional exponential averager (also called a "leaky integrator"), shown in Figure 1(a), is commonly used to reduce noise fluctuations that contaminate relatively constant-amplitude signal measurements.
Figure 1 Exponential averaging: (a) standard network; (b) single-multiply network.That exponential averager's difference equation is
y(n) = αx(n) + (1 –...
