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...
"Neat" Rectangular to Polar Conversion Algorithm
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 (Mr. Wizard) Avins posted an interesting message on this subject to the comp.dsp newsgroup (Subject: "Re:
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...
Computing FFT Twiddle Factors
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 algorithms showing how to compute the individual twiddle factors of an N-point decimation-in-frequency...
Computing an FFT of Complex-Valued Data Using a Real-Only FFT Algorithm
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 to this problem. Here is my idea:
Let's say our original input data is the complex-valued sequence...
Some Thoughts on a German Mathematician
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 seventeen years?) Gauss was so prolific that by some estimates he personally doubled the amount of...
Using Mason's Rule to Analyze DSP Networks
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 valuable analysis method (well known to our analog control system engineering brethren) to obtain the z-domain...
Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT
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 seven steps, S1 through S7, shown in Figure 1. In that figure, we compute the x(n) inverse FFT of...
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 –...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...A DSP Quiz Question
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 in Figure 1(b).
HERE'S THE QUIZ QUESTION:
I was reading a paper by an audio DSP engineer where the...Microprocessor Family Tree
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.
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...
A Useful Source of Signal Processing Information
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 quotation marks) into the 'Search' box at the top of the web page
and click the red 'SEARCH...
The Swiss Army Knife of Digital Networks
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 series of examples I've illustrated the fundamental strength of this Swiss Army Knife of digital networks...
The Little Fruit Market: The Beginning of the Digital Explosion
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 physicist William Shockley, along with John Bardeen and Walter Brattain, co-invented the transistor at Bell...
A Remarkable Bit of DFT Trivia
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 of the sine wave's positive-frequency spectral component will be
where A is the peak amplitude...
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)...
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...
Implementing Impractical Digital Filters
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 practical.
Implementing an Impractical Filter: Example 1
Reference [1] presented the digital IIR bandpass filter...
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...A Useful Source of Signal Processing Information
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 quotation marks) into the 'Search' box at the top of the web page
and click the red 'SEARCH...
Implementing Simultaneous Digital Differentiation, Hilbert Transformation, and Half-Band Filtering
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.
The Risk In Using Frequency Domain Curves To Evaluate Digital Integrator Performance
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 closely that integrator's frequency response matches the frequency response of an ideal integrator [1,2]....
"Neat" Rectangular to Polar Conversion Algorithm
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 (Mr. Wizard) Avins posted an interesting message on this subject to the comp.dsp newsgroup (Subject: "Re:
Controlling a DSP Network's Gain: A Note For DSP Beginners
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.
FIGURE 1. Examples of digital networks whose initial operations are input signal...
A Wide-Notch Comb Filter
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 1(c) shows the filter's positive-frequency magnitude response when, for example, D = 9. The...A Table of Digital Frequency Notation
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 the units of measure (dimensions) of those variables are different. It's sometimes a nuisance to keep...
Simultaneously Computing a Forward FFT and an Inverse FFT Using a Single FFT
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 seven steps, S1 through S7, shown in Figure 1. In that figure, we compute the x(n) inverse FFT of...
