A Fast Guaranteed-Stable Sliding DFT Algorithm

Rick Lyons — Thu, 15 Jun 2023 17:41:19 +0000

This blog presents a most computationally-efficient guaranteed-stable real-time sliding discrete Fourier transform (SDFT) algorithm. The phrase “real-time” means the network computes one spectral output sample, equal to a single-bin output of an N‑point discrete Fourier transform (DFT), for each input signal sample.

Proposed Guaranteed Stable SDFT

My proposed guaranteed stable

A New Contender in the Quadrature Oscillator Race

Rick Lyons — Sat, 24 Sep 2022 22:42:25 +0000

This blog advocates a relatively new and interesting quadrature oscillator developed by A. David Levine in 2009 and independently by Martin Vicanek in 2015 [1]. That oscillator is shown in Figure 1.

The time domain equations describing the Figure 1 oscillator are

w(n) =...

A DSP Quiz Question

Rick Lyons — Sun, 05 Dec 2021 20:55:37 +0000

Here's a DSP Quiz Question that I hope you find mildly interesting

BACKGROUND

Due to the periodic natures an N-point discrete Fourier transform (DFT) sequence and that sequence’s inverse DFT, it is occasionally reasonable to graphically plot either of those sequences as a 3-dimensional (3D) circular plot. For example, Figure 1(a) shows a length-32 x(n) sequence with its 3D circular plot given...

An Efficient Full-Band Sliding DFT Spectrum Analyzer

Rick Lyons — Thu, 01 Apr 2021 18:15:41 +0000

In this blog I present two computationally efficient full-band discrete Fourier transform (DFT) networks that compute the 0th bin and all the positive-frequency bin outputs for an N-point DFT in real-time on a sample-by-sample basis.

An Even-N Spectrum Analyzer

The full-band sliding DFT (SDFT) spectrum analyzer network, where the DFT size N is an even integer, is shown in Figure 1(a). The...

A Simpler Goertzel Algorithm

Rick Lyons — Thu, 04 Feb 2021 09:41:28 +0000

In this blog I propose a Goertzel algorithm that is simpler than the version of the Goertzel algorithm that is traditionally presented DSP textbooks. Below I very briefly describe the DSP textbook version of the Goertzel algorithm followed by a description of my proposed simpler algorithm.

The Traditional DSP Textbook Goertzel Algorithm

The so-called Goertzel algorithm is used to...

60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing

Rick Lyons — Sat, 23 Jan 2021 09:31:32 +0000

Electrocardiogram (ECG) signals are obtained by monitoring the electrical activity of the human heart for medical diagnostic purposes [1]. This blog describes a very efficient digital filter used to reduce both 60 Hz AC power line noise and unwanted signal baseline drift that often contaminate ECG signals.

PDF_HERE

We'll first describe the ECG noise reduction filter and then examine the...

A Fast Real-Time Trapezoidal Rule Integrator

Rick Lyons — Sat, 13 Jun 2020 21:20:08 +0000

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...

A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters

Rick Lyons — Thu, 26 Mar 2020 17:54:09 +0000

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...

The DFT of Finite-Length Time-Reversed Sequences

Rick Lyons — Fri, 20 Dec 2019 13:40:16 +0000

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...

Update To: A Wide-Notch Comb Filter

Rick Lyons — Mon, 09 Dec 2019 20:16:59 +0000

This blog presents alternatives to the wide-notch comb filter described in Reference [1]. That comb filter, which for notational reasons I now call a 2-RRS wide notch comb filter, is shown in Figure 1. I use the "2-RRS" moniker because the comb filter uses two recursive running sum (RRS) networks.

The z-domain transfer function of the 2-RRS wide-notch comb filter, H2-RRS(z), is:

...

A Wide-Notch Comb Filter

Rick Lyons — Sun, 24 Nov 2019 14:06:14 +0000

This blog describes a linear-phase comb filter having wider stopband notches than a traditional comb filter.

Background

Let's first review the behavior of a traditional comb filter. Figure 1(a) shows a traditional comb filter comprising two cascaded recursive running sum (RRS) comb filters. Figure 1(b) shows the filter's co-located dual poles and dual zeros on the z-plane, while Figure...

The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance

Rick Lyons — Tue, 24 Sep 2019 09:55:15 +0000

This blog shows the danger in evaluating the performance of a digital integration network based solely on its frequency response curve. If you plan on implementing a digital integrator in your signal processing work I recommend you continue reading this blog.

Background

Typically when DSP practitioners want to predict the accuracy performance of a digital integrator they compare how...

Reduced-Delay IIR Filters

Rick Lyons — Thu, 04 Jul 2019 16:52:54 +0000

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...

Somewhat Off Topic: Deciphering Transistor Terminology

Rick Lyons — Wed, 29 May 2019 00:51:42 +0000

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...

Reducing IIR Filter Computational Workload

Rick Lyons — Fri, 24 May 2019 16:58:01 +0000

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...

A Lesson In Engineering Humility

Rick Lyons — Mon, 20 May 2019 16:55:07 +0000

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.

Controlling a DSP Network's Gain: A Note For DSP Beginners

Rick Lyons — Fri, 29 Mar 2019 04:16:14 +0000

This blog briefly discusses a topic well-known to experienced DSP practitioners but may not be so well-known to DSP beginners. The topic is the proper way to control a digital network's gain. Digital Network Gain Control Figure 1 shows a collection of networks I've seen, in the literature of DSP, where strict gain control is implemented.

...

Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'

Rick Lyons — Wed, 20 Feb 2019 17:51:00 +0000

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...

A Brief Introduction To Romberg Integration

Rick Lyons — Wed, 16 Jan 2019 14:17:45 +0000

This blog briefly describes a remarkable integration algorithm, called "Romberg integration." The algorithm is used in the field of numerical analysis but it's not so well-known in the world of DSP.

To show the power of Romberg integration, and to convince you to continue reading, consider the notion of estimating the area under the continuous x(t) = sin(t) curve based on the five...

Microprocessor Family Tree

Rick Lyons — Thu, 10 Jan 2019 22:29:35 +0000

Below is a little microprocessor history. Perhaps some of the ol' timers here will recognize a few of these integrated circuits. I have a special place in my heart for the Intel 8080 chip.

Image copied, without permission, from the now defunct Creative Computing magazine, Vol. 11, No. 6, June 1985.

Two Easy Ways To Test Multistage CIC Decimation Filters

Rick Lyons — Tue, 22 May 2018 18:13:47 +0000

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...

FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending

Rick Lyons — Mon, 16 Apr 2018 19:33:09 +0000

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...

An Efficient Linear Interpolation Scheme

Rick Lyons — Wed, 27 Dec 2017 13:29:45 +0000

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.

Errata 3rd Ed. International Version.pdfErrata 3rd Ed. International Version.pdf

This 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...

Above-Average Smoothing of Impulsive Noise

Rick Lyons — Mon, 10 Jul 2017 17:32:07 +0000

In this blog I show a neat noise reduction scheme that has the high-frequency noise reduction behavior of a traditional moving average process but with much better impulsive-noise suppression.

In practice we may be required to make precise measurements in the presence of highly-impulsive noise. Without some sort of analog signal conditioning, or digital signal processing, it can be...

Looking For a Second Toolbox? This One's For Sale

Rick Lyons — Thu, 29 Jun 2017 13:46:12 +0000

In case you're looking for a second toolbox, this used toolbox is for sale.

The blue-enameled steel toolbox measures 13 x 7 x 5 inches and, when opened, has a three-section tray attached to the lid. Showing signs of heavy use, the interior, tray, and exterior have collected a fair amount of dirt and grease and bear many scratches. The bottom of the box is worn from having been slid on...

Sinusoidal Frequency Estimation Based on Time-Domain Samples

Rick Lyons — Thu, 20 Apr 2017 04:41:51 +0000

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...

Frequency Translation by Way of Lowpass FIR Filtering

Rick Lyons — Sat, 04 Feb 2017 19:10:48 +0000

Some weeks ago a question appeared on the dsp.related Forum regarding the notion of translating a signal down in frequency and lowpass filtering in a single operation [1]. It is possible to implement such a process by embedding a discrete cosine sequence's values within the coefficients of a traditional lowpass FIR filter. I first learned about this process from Reference [2]. Here's the...

The Real Star of Star Trek

Rick Lyons — Sun, 25 Sep 2016 19:26:16 +0000

Unless you've been living under a rock recently, you're probably aware that this month is the 50-year anniversary of the original Star Trek show on American television. It's an anniversary worth noting, as did Time and Newsweek magazines with their special editions.

Over the years I've come to realize that a major star of the original Star Trek series wasn't an actor....

An s-Plane to z-Plane Mapping Example

Rick Lyons — Sat, 24 Sep 2016 11:51:14 +0000

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...

Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Rick Lyons — Wed, 14 Sep 2016 13:47:27 +0000

Not long ago I presented my 3-day DSP class to a group of engineers at Tektronix Inc. in Beaverton Oregon [1]. After I finished covering my material on IIR filters' z-plane pole locations and filter stability, one of the Tektronix engineers asked a question similar to:

"I noticed that you didn't discuss z-plane regions of convergence here. In my...

Implementing Impractical Digital Filters

Rick Lyons — Tue, 19 Jul 2016 13:29:08 +0000

This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them...

An Astounding Digital Filter Design Application

Rick Lyons — Thu, 07 Jul 2016 19:57:09 +0000

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...

The Swiss Army Knife of Digital Networks

Rick Lyons — Mon, 13 Jun 2016 19:56:48 +0000

This blog describes a general discrete-signal network that appears, in various forms, inside so many DSP applications.

Figure 1 shows how the network's structure has the distinct look of a digital filter—a comb filter followed by a 2nd-order recursive network. However, I do not call this useful network a filter because its capabilities extend far beyond simple filtering. Through a...

Digital Envelope Detection: The Good, the Bad, and the Ugly

Rick Lyons — Sun, 03 Apr 2016 14:38:15 +0000

Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.

Here I focus on envelope detection as it is applied to an...

A Useful Source of Signal Processing Information

Rick Lyons — Wed, 23 Mar 2016 16:09:46 +0000

I just discovered a useful web-based source of signal processing information that was new to me. I thought I'd share what I learned with the subscribers here on DSPRelated.com.

The Home page of the web site that I found doesn't look at all like it would be useful to us DSP fanatics. But if you enter some signal processing topic of interest, say, "FM demodulation" (without the...

Optimizing the Half-band Filters in Multistage Decimation and Interpolation

Rick Lyons — Mon, 04 Jan 2016 19:59:52 +0000

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...

Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering

Rick Lyons — Tue, 24 Nov 2015 14:42:07 +0000

Recently I've been thinking about digital differentiator and Hilbert transformer implementations and I've developed a processing scheme that may be of interest to the readers here on dsprelated.com.

This blog presents a novel method for simultaneously implementing a digital differentiator (DD), a Hilbert transformer (HT), and a half-band lowpass filter (HBF) using a single tapped-delay line...

A New Contender in the Digital Differentiator Race

Rick Lyons — Wed, 30 Sep 2015 13:25:46 +0000

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.

Background

In reference [1] I presented a computationally-efficient tapped-delay line digital differentiator whose $h_{ref}(k)$ impulse...

The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase

Rick Lyons — Tue, 18 Aug 2015 16:33:40 +0000

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

Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm

Rick Lyons — Tue, 07 Jul 2015 20:58:57 +0000

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}...

Correcting an Important Goertzel Filter Misconception

Rick Lyons — Mon, 06 Jul 2015 15:56:55 +0000

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...

Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling

Rick Lyons — Mon, 04 May 2015 13:13:44 +0000

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,...

Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

Rick Lyons — Mon, 23 Mar 2015 13:42:20 +0000

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...

Complex Down-Conversion Amplitude Loss

Rick Lyons — Tue, 03 Mar 2015 14:37:22 +0000

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...

A Complex Variable Detective Story – A Disconnect Between Theory and Implementation

Rick Lyons — Tue, 14 Oct 2014 18:54:50 +0000

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...

The Number 9, Not So Magic After All

Rick Lyons — Thu, 02 Oct 2014 01:28:49 +0000

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...

Sum of Two Equal-Frequency Sinusoids

Rick Lyons — Thu, 04 Sep 2014 18:18:04 +0000

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...

The DFT Magnitude of a Real-valued Cosine Sequence

Rick Lyons — Tue, 17 Jun 2014 14:23:16 +0000

This blog may seem a bit trivial to some readers here but, then again, it might be of some value to DSP beginners. It presents a mathematical proof of what is the magnitude of an N-point discrete Fourier transform (DFT) when the DFT's input is a real-valued sinusoidal sequence.

To be specific, if we perform an N-point DFT on N real-valued time-domain samples of a discrete cosine wave, having...

Specifying the Maximum Amplifier Noise When Driving an ADC

Rick Lyons — Mon, 09 Jun 2014 19:47:38 +0000

I recently learned an interesting rule of thumb regarding the use of an amplifier to drive the input of an analog to digital converter (ADC). The rule of thumb describes how to specify the maximum allowable noise power of the amplifier [1].

The Problem Here's the situation for an ADC whose maximum analog input voltage range is –VRef to +VRef. If we drive an ADC's analog input with an sine...

A Remarkable Bit of DFT Trivia

Rick Lyons — Thu, 26 Dec 2013 16:43:09 +0000

I recently noticed a rather peculiar example of discrete Fourier transform (DFT) trivia; an unexpected coincidence regarding the scalloping loss of the DFT. Here's the story.

DFT SCALLOPING LOSS As you know, if we perform an N-point DFT on N real-valued time-domain samples of a discrete sine wave, whose frequency is an integer multiple of fs/N (fs is the sample rate in Hz), the peak magnitude...

Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals

Rick Lyons — Thu, 31 Oct 2013 17:54:32 +0000

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...

Goertzel Algorithm for a Non-integer Frequency Index

Rick Lyons — Mon, 07 Oct 2013 14:23:37 +0000

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...

Is It True That j is Equal to the Square Root of -1 ?

Rick Lyons — Mon, 16 Sep 2013 18:48:39 +0000

A few days ago, on the YouTube.com web site, I watched an interesting video concerning complex numbers and the j operator. The video's author claimed that the statement "j is equal to the square root of negative one" is incorrect. What he said was:

He justified his claim by going through the following exercise, starting with:

Based on the algebraic...

A Table of Digital Frequency Notation

Rick Lyons — Mon, 05 Aug 2013 15:50:06 +0000

When we read the literature of digital signal processing (DSP) we encounter a number of different, and equally valid, ways to algebraically represent the notion of frequency for discrete-time signals. (By frequency I mean a measure of angular repetitions per unit of time.)

The various mathematical expressions for sinusoidal signals use a number of different forms of a frequency variable and...

A Quadrature Signals Tutorial: Complex, But Not Complicated

Rick Lyons — Fri, 12 Apr 2013 13:19:56 +0000

Introduction Quadrature signals are based on the notion of complex numbers and perhaps no other topic causes more heartache for newcomers to DSP than these numbers and their strange terminology of j operator, complex, imaginary, real, and orthogonal. If you're a little unsure of the physical meaning of complex numbers and the j = √-1 operator, don't feel bad because you're in good company....

Beat Notes: An Interesting Observation

Rick Lyons — Wed, 13 Mar 2013 14:25:33 +0000

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...

Using the DFT as a Filter: Correcting a Misconception

Rick Lyons — Mon, 18 Feb 2013 19:11:42 +0000

I have read, in some of the literature of DSP, that when the discrete Fourier transform (DFT) is used as a filter the process of performing a DFT causes an input signal's spectrum to be frequency translated down to zero Hz (DC). I can understand why someone might say that, but I challenge that statement as being incorrect. Here are my thoughts.

Using the DFT as a Filter It may seem strange to...

The Little Fruit Market: The Beginning of the Digital Explosion

Rick Lyons — Mon, 14 Jan 2013 14:19:46 +0000

There used to be a fruit market located at 391 San Antonio Road in Mountain View, California. In the 1990's I worked part time in Mountain View and drove past this market's building, shown in Figure 1, many times, unaware of its history. What happened at that fruit market has changed the lives of almost everyone on our planet. Here's the story.

William Shockley In 1948 the brilliant...

Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved

Rick Lyons — Fri, 23 Nov 2012 15:49:55 +0000

This blog clarifies how to obtain and interpret the z-domain transfer function of the coupled-form 2nd-order IIR resonator. The coupled-form 2nd-order IIR resonator was developed to overcome a shortcoming in the standard 2nd-order IIR resonator. With that thought in mind, let's take a brief look at a standard 2nd-order IIR resonator.

Standard 2nd-Order IIR Resonator A block diagram of the...

Setting the 3-dB Cutoff Frequency of an Exponential Averager

Rick Lyons — Mon, 22 Oct 2012 15:02:15 +0000

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...

Understanding the 'Phasing Method' of Single Sideband Demodulation

Rick Lyons — Wed, 08 Aug 2012 13:28:11 +0000

There are four ways to demodulate a transmitted single sideband (SSB) signal. Those four methods are:

synchronous detection,
phasing method,
Weaver method, and
filtering method.

Here we review synchronous detection in preparation for explaining, in detail, how the phasing method works. This blog contains lots of preliminary information, so if you're already familiar with SSB...

How Discrete Signal Interpolation Improves D/A Conversion

Rick Lyons — Mon, 28 May 2012 12:17:47 +0000

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...

How Not to Reduce DFT Leakage

Rick Lyons — Wed, 23 May 2012 18:49:21 +0000

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

The History of CIC Filters: The Untold Story

Rick Lyons — Mon, 20 Feb 2012 13:52:38 +0000

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...

Accurate Measurement of a Sinusoid's Peak Amplitude Based on FFT Data

Rick Lyons — Wed, 14 Dec 2011 13:56:48 +0000

There are two code snippets associated with this blog post:

Flat-Top Windowing Function for the Accurate Measurement of a Sinusoid's Peak Amplitude Based on FFT Data

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...

Generating Complex Baseband and Analytic Bandpass Signals

Rick Lyons — Wed, 02 Nov 2011 13:32:26 +0000

There are so many different time- and frequency-domain methods for generating complex baseband and analytic bandpass signals that I had trouble keeping those techniques straight in my mind. Thus, for my own benefit, I created a kind of reference table showing those methods. I present that table for your viewing pleasure in this blog.

For clarity, I define a complex baseband signal as...

Orfanidis Textbooks are Available Online

Rick Lyons — Tue, 12 Jul 2011 12:40:28 +0000

I have just learned that Sophocles J. Orfanidis, the well-known professor with the ECE Department of Rutgers University, has made two of his signal processing textbooks available for downloading on the Internet. The first textbook is: "Introduction to Signal Processing" available at: http://eceweb1.rutgers.edu/~orfanidi/intro2sp/

Happily, also available at the above web site...

Do Multirate Systems Have Transfer Functions?

Rick Lyons — Mon, 30 May 2011 15:01:22 +0000

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...

Multiplying Two Binary Numbers

Rick Lyons — Wed, 16 Mar 2011 23:08:21 +0000

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...

"Neat" Rectangular to Polar Conversion Algorithm

Rick Lyons — Mon, 15 Nov 2010 19:07:29 +0000

The subject of finding algorithms that estimate the magnitude of a complex number, without having to perform one of those pesky square root operations, has been discussed many times in the past on the comp.dsp newsgroup. That is, given the complex number R + jI in rectangular notation, we want to estimate the magnitude of that number represented by M as:

On August 25th, 2009, Jerry...

Improved Narrowband Lowpass IIR Filters

Rick Lyons — Sat, 06 Nov 2010 12:43:10 +0000

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...

Computing FFT Twiddle Factors

Rick Lyons — Sun, 08 Aug 2010 03:16:36 +0000

Some days ago I read a post on the comp.dsp newsgroup and, if I understood the poster's words, it seemed that the poster would benefit from knowing how to compute the twiddle factors of a radix-2 fast Fourier transform (FFT).

Then, later it occurred to me that it might be useful for this blog's readers to be aware of algorithms for computing FFT twiddle factors. So,... what follows are two...

Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm

Rick Lyons — Tue, 09 Feb 2010 16:40:04 +0000

Someone recently asked me if I knew of a way to compute a fast Fourier transform (FFT) of complex-valued input samples using an FFT algorithm that accepts only real-valued input data. Knowing of no way to do this, I rifled through my library of hardcopy FFT articles looking for help. I found nothing useful that could be applied to this problem.

After some thinking, I believe I have a solution...

Some Thoughts on a German Mathematician

Rick Lyons — Mon, 11 Jan 2010 16:49:48 +0000

Carl Friedrich Gauss

Here are a few interesting facts about the great Carl Friedrich Gauss (1777-1855), considered by some historians to have been the world's greatest mathematician. The overused phrase of "genius" could, with full justification, be used to describe this man. (How many people do you know that could have discovered the law of quadratic reciprocity in number theory at the age...

Using Mason's Rule to Analyze DSP Networks

Rick Lyons — Mon, 31 Aug 2009 23:06:18 +0000

There have been times when I wanted to determine the z-domain transfer function of some discrete network, but my algebra skills failed me. Some time ago I learned Mason's Rule, which helped me solve my problems. If you're willing to learn the steps in using Mason's Rule, it has the power of George Foreman's right hand in solving network analysis problems.

This blog discusses a...

Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT

Rick Lyons — Tue, 13 Jan 2009 17:44:13 +0000

Most of us are familiar with the processes of using a single N-point complex FFT to: (1) perform a 2N-point FFT on real data, and (2) perform two independent N-point FFTs on real data [1–5]. In case it's of interest to someone out there, this blog gives the algorithm for simultaneously computing a forward FFT and an inverse FFT using a single radix-2 FFT.

Our algorithm is depicted by the...

Multiplierless Exponential Averaging

Rick Lyons — Fri, 05 Dec 2008 15:11:40 +0000

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...

Free DSP Books on the Internet - Part Deux

Rick Lyons — Thu, 04 Dec 2008 13:06:33 +0000

Since Stephane Boucher posted my "Free DSP Books on the Internet" blog here in February 2008, I have learned of additional books on the Internet that are related to signal processing. I list those books below. Again, the listed books are copyrighted. The books' copyright holders have graciously provided their books free of charge for downloading for individual use, but multiple copies must...

Computing the Group Delay of a Filter

Rick Lyons — Wed, 19 Nov 2008 13:15:14 +0000

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...

Computing Large DFTs Using Small FFTs

Rick Lyons — Tue, 24 Jun 2008 01:45:41 +0000

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...

Linear-phase DC Removal Filter

Rick Lyons — Sun, 30 Mar 2008 11:04:20 +0000

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...

Free DSP Books on the Internet

Rick Lyons — Sun, 24 Feb 2008 01:04:57 +0000

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...

A Simple Complex Down-conversion Scheme

Rick Lyons — Mon, 21 Jan 2008 14:03:53 +0000

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...

Computing Chebyshev Window Sequences

Rick Lyons — Tue, 08 Jan 2008 04:12:57 +0000

Chebyshev windows (also called Dolph-Chebyshev, or Tchebyschev windows), have several useful properties. Those windows, unlike the fixed Hanning, Hamming, or Blackman window functions, have adjustable sidelobe levels. For a given user-defined sidelobe level and window sequence length, Chebyshev windows yield the most narrow mainlobe compared to any fixed window functions.

Spectral Flipping Around Signal Center Frequency

Rick Lyons — Thu, 08 Nov 2007 01:41:32 +0000

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...

A Differentiator With a Difference

Rick Lyons — Sat, 03 Nov 2007 10:25:05 +0000

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...

Rick Lyons Blog on DSPRelated.com

Algebra's Laws of Powers and Roots: Handle With Care

A Fast Guaranteed-Stable Sliding DFT Algorithm

A New Contender in the Quadrature Oscillator Race

A DSP Quiz Question

An Efficient Full-Band Sliding DFT Spectrum Analyzer

A Simpler Goertzel Algorithm

60-Hz Noise and Baseline Drift Reduction in ECG Signal Processing

A Fast Real-Time Trapezoidal Rule Integrator

A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters

The DFT of Finite-Length Time-Reversed Sequences

Update To: A Wide-Notch Comb Filter

A Wide-Notch Comb Filter

The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance

Reduced-Delay IIR Filters

Somewhat Off Topic: Deciphering Transistor Terminology

Reducing IIR Filter Computational Workload

A Lesson In Engineering Humility

Controlling a DSP Network's Gain: A Note For DSP Beginners

Stereophonic Amplitude-Panning: A Derivation of the 'Tangent Law'

A Brief Introduction To Romberg Integration

Microprocessor Family Tree

Two Easy Ways To Test Multistage CIC Decimation Filters

FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending

An Efficient Linear Interpolation Scheme

Above-Average Smoothing of Impulsive Noise

Looking For a Second Toolbox? This One's For Sale

Sinusoidal Frequency Estimation Based on Time-Domain Samples

Frequency Translation by Way of Lowpass FIR Filtering

The Real Star of Star Trek

An s-Plane to z-Plane Mapping Example

Should DSP Undergraduate Students Study z-Transform Regions of Convergence?

Implementing Impractical Digital Filters

An Astounding Digital Filter Design Application

The Swiss Army Knife of Digital Networks

Digital Envelope Detection: The Good, the Bad, and the Ugly

A Useful Source of Signal Processing Information

Optimizing the Half-band Filters in Multistage Decimation and Interpolation

Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering

A New Contender in the Digital Differentiator Race

The Most Interesting FIR Filter Equation in the World: Why FIR Filters Can Be Linear Phase

Four Ways to Compute an Inverse FFT Using the Forward FFT Algorithm

Correcting an Important Goertzel Filter Misconception

Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling

Why Time-Domain Zero Stuffing Produces Multiple Frequency-Domain Spectral Images

Complex Down-Conversion Amplitude Loss

A Complex Variable Detective Story – A Disconnect Between Theory and Implementation

The Number 9, Not So Magic After All

Sum of Two Equal-Frequency Sinusoids

The DFT Magnitude of a Real-valued Cosine Sequence

Specifying the Maximum Amplifier Noise When Driving an ADC

A Remarkable Bit of DFT Trivia

Computing Translated Frequencies in Digitizing and Downsampling Analog Bandpass Signals

Goertzel Algorithm for a Non-integer Frequency Index

Is It True That j is Equal to the Square Root of -1 ?

A Table of Digital Frequency Notation

A Quadrature Signals Tutorial: Complex, But Not Complicated

Beat Notes: An Interesting Observation

Using the DFT as a Filter: Correcting a Misconception

The Little Fruit Market: The Beginning of the Digital Explosion

Coupled-Form 2nd-Order IIR Resonators: A Contradiction Resolved

Setting the 3-dB Cutoff Frequency of an Exponential Averager

Understanding the 'Phasing Method' of Single Sideband Demodulation

How Discrete Signal Interpolation Improves D/A Conversion

How Not to Reduce DFT Leakage

The History of CIC Filters: The Untold Story

Accurate Measurement of a Sinusoid's Peak Amplitude Based on FFT Data

Generating Complex Baseband and Analytic Bandpass Signals

Orfanidis Textbooks are Available Online

Do Multirate Systems Have Transfer Functions?

Multiplying Two Binary Numbers

"Neat" Rectangular to Polar Conversion Algorithm

Improved Narrowband Lowpass IIR Filters

Computing FFT Twiddle Factors

Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm

Some Thoughts on a German Mathematician

Using Mason's Rule to Analyze DSP Networks

Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT

Multiplierless Exponential Averaging

Free DSP Books on the Internet - Part Deux