**The Intuitive Guide to
**

Available at Amazon.com as color eBook to be read on your PC or Ipad.

320 pages

Print hardcover version will be available Dec 2016

For Matlab code – Please check back. Site currently in construction.

Questions and Answers – Please check back. Site currently in construction.

One can get a pretty good feeling about a book by just flipping a few pages. And this book passed the flip test immediately!

First of all, I want to say that I love this book and the way the material is presented, from simplest to the complete and comprehensive. It covers the usual course material of a fist DSP/signal processing class. The first chapter starts with looking at common signals, such sinusoids, exponentials etc. but presents then not just by an equation by many graphs and pictures with emphasis on visual understanding. Matlab code that produced these figures is included and teaches these in a way to help the student to further their signal manipulation skills. Then end of chapter exercises are concise and actually helpful and not an IQ test. What is remarkable about this book is that it often includes examples of a real life process of how a signal might have been created, such as it does for the half-wave rectifier giving a circuit realization. This book does continually through all chapters and is unique for DSP book. In the linear systems chapter, the author goes much further than most in helping through graphs, the understanding of convolution. There are probably 20 pages on convolution a subject which because of ridiculous simplicity so confuses all of us.

The Authors then cover the Fourier series, giving once again examples such as the 60 Hz powerline, the CB Radio booster, and other amplifiers are examples of how Fourier analysis is applied. I am myself writing a book on Fourier transform so I was pleased that they have an approach similar to mine in not confusing but making the topic crystal clear.

The Laplace Transform is made clear with several examples and end of chapter examples are actually help bring the essence home. I could go on and on for each chapter. The quality never varies and remains high, all topics are covered well. The last chapter on communications brings the applications of the fundamental concepts all in one place and is an excellent summary.

Excellent textbook or a brush up on concepts for a working engineer.

Practical Signal Theory with Matlab Applications, Richard J. Tervo, John Wiley & Sons Inc. 2014

________________

*I received this book from the publisher for review. I don’t receive that many and this one is actually the first.*

]]>

The book will have some of my DFT, FFT material, plus Matlab code. It will also have chapters on spectral estimation which is ultimately the reason why we do the FFTs.

The DTFT we all know and love is mathematically valid only for deterministic signals. Real life signals are random, even when they have some deterministic components, such as a carrier. Per Dirichlet, we can not just take the DTFT of a random signal. That is a no-no. What we need to do is to take the auto-correlation first and then the DTFT of the Auto-Correlation Function (ACF). There are the issues of windowing and the length of the ACF. Then we have parametric and non-parametric estimation methods. Spectral estimation is a complex topic and there is little agreement on how to do it right. There are numerous complexities in understanding what the hardware is producing, how to best match it in simulation, and, most importantly, if the results are valid. I got carried away writing my last chapter on spectral estimation and even then never got to the parametric estimation which is a whole another world of mathematics, used mostly for describing finance, environmental, and biological phenomena.

I am working on this book with my son Victor as a co-author. He had just completed his masters in EE at Georgia Tech and is looking for a job in wireless or embedded systems. If anyone is looking for a smart young engineer, here is a link to his page.

Charan Langton

]]>**by Richard Lyons**

Some time ago Charan asked me what events led me to write my “Understanding Digital Signal Processing” book? For Charan, and in case anyone else is interested, here’s the story.

In the mid-1980s I wanted to learn digital signal processing (DSP). As I tried to learn DSP it was the best of times, it was the worst of times. It was a period of understanding, it was a period of confusion, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, I had everything before me, I had nothing before me, I was approaching enlightenment, I was doomed to ignorance. There were kings of DSP with large jaws and mathematical minds on their thrones in universities bestowing their knowledge in cryptic form. In their ivory castles, it was clearer than crystal to the lords of technology that things were settled forever.

With thanks, and apologies, to Charles Dickens, I’ll stop “clowning around” and just say that decades ago was a dismal time to try to learn DSP on one’s own. The available DSP textbooks in the mid 1980’s were, for practical purposes, unreadable.

Back then, written explanations of DSP theory appeared in one of two forms: (1) mathematical miracles occur where you’re simply given a short-n-sweet equation without further explanation; or (2) you faced a flood of complex variable equations and phrases such as “it is obvious that”, “such that W(f)≥ Åf”, and “with judicious application of the homogeneity property.” In their defense, those DSP authors provide the needed information, but too often the reader had to grab a pick and shovel, put on a miner’s helmet, and try to dig the information out of a mountain of mathematical expressions.

How many times have you been forced to follow the derivation of an equation, after which the author states they’re going to illustrate that equation with a physical example, which turns out to be just another equation? Anyway, in the mid-1980s I needed help, other than cryptic textbooks, to learn DSP.

If you’re still with me, here’s why I wrote my DSP book. In 1987 I took an evening community college course on the subject of DSP. The class textbook wasn’t too awful bad. After reading its material 2-3 times, some of its DSP theory began to slowly sink into my head. The second half of the textbook was a series of applications notes from various hardware vendors. (I suppose the co-authors needed some “filler” material to make the page count of their book acceptable to their publisher.) In any case, I happened to find a significant conceptual error in one of those application notes with regard to the topic of periodic sampling. You see, at that time I had spent months on my own studying periodic sampling. It was the only DSP topic that I understood at the time. (In 1988 I wrote a ‘periodic sampling’ article for the “Test and Measurement” magazine.)

Finding that major error in the textbook was a pivotal moment for me. It made me realize that textbook authors aren’t Gods after all! Maybe they’re smarter than me, and maybe they aren’t. If they’re so smart, why didn’t *they* catch that important periodic sampling error?

Well, …in 1990 I had a wild idea and wrote a letter to the primary author of my evening DSP class textbook, a professor at a university on the east coast of the United States. I volunteered to write a chapter on periodic sampling for the next edition of his book, assuming there might be a new edition.

In my letter to the professor I explained my idea. He requested my periodic sampling material and assured me that he’d review it and not plagiarize any of it. Cautiously excited, I thought, “Hey Rick, if you work hard enough you might actually get your name listed as a co-author of a book.”

After refining, and expanding, my periodic sampling material I mailed it to the professor. I waited and waited for a reply. Well, I ***NEVER*** heard from the good professor again! He returned none of my phone calls nor answered any of my subsequent letters. (There was no such thing as e-mail at that time.) In early 1991 I decided, “Heck with the professor, I’ll write my own book.” So I did.

To answer the question in the title of this blog, I wrote my first DSP book out of an untested, and unsupported, optimism that I could explain the basic concepts of DSP better than the DSP books available in 1991. I finished writing the book’s first edition and submitted that manuscript to my publisher in 1996. (In case you’re doing the math, yes, it took me five years to produce that first edition.)

Two years later, in late 1998, I found the east-coast professor’s E-mail address on the Internet. Being my cantankerous self, I sent him the following E-mail thanking him for ignoring me all those years ago. (Out of common decency I won’t give the professor’s name here. If you want to know his name, contact the NSA. I’ll bet they have a copy of my e-mail.) Here’s the content of my e-mail:

From: rick lyons To: Professor@xxx.edu

Date: 12/10/98 08:16:19

Dear Professor,

I owe you a debt of gratitude. In May of 1990 I wrote you a letter

in which I volunteered to contribute some material to the (at that

time) next edition of your digital signal processing textbook. You

answered my letter and encouraged me to forward my material (on

periodic sampling) for your review, and you assured me that you’d

not plagiarize any of that material. I did mail my periodic sampling

text and figures to you in late May, 1990. After that, you failed

to return any of my phone calls, or respond to any of my subsequent

letters. This disappointed and annoyed me so much that I became

stubborn and decided to write my own DSP book.

My book (“Understanding Digital Signal Processing”) was published a

few years ago and has been surprisingly successful. So successful, in

fact, that it’s changed my life for the better. Had you answered

my phone calls or letters, I’d still be working 45 hours per

week for some aerospace company. Now I spend my time as a private

contractor, lecturer, and author – and for this, I thank you very much.

Regards,

[-Richard Lyons-]

This time, the professor replied to me the same day(!) with:

Date: Thu, 10 Dec 1998 18:37:00

To: rick lyons

From: the professor

Subject: Re: Thank You

Dear Richard:

Thank you for the email and note about your book “Understanding

Digital Signal Processing”. …I am glad this has worked out

very well for you. However, I am one of those sensitive people

who goes out of his way to try not to offend others, so I would

like to clarify the situation of some years back.

At the time you contacted me, I was swamped with work and was not

in a position to revamp the DSP book. I definitely remember having

several conversations with you and receiving the material you sent

(which has never gone any further than my office). If I failed to

respond to you, please accept my apologies since there certainly

no intent to be insensitive to you. My failure to respond was

simply a “statement” that I was not in a position at the time to

do any more work in the area.

Again, accept my apologies, and congratulations on what sounds

like an excellent career move!

Sincerely,

The professor’s reply was fairly gracious, I must say.

To the professor, I must confess, I painted an overly-rosy picture of the situation in which I found myself after resigning from my full-time aerospace job in 1998. Indeed, all I had to do to survive after my resignation was not buy anything, not go anywhere, have no hobbies or personal life, and accept Government cheese to eat.

In case anyone reading this blog has any bright book-writing ideas, know this. It’s not possible to make a living off the royalties of a signal processing book. To make money writing a book, your book had better contain massive amounts of senseless violence and meaningless sex.

Rick Lyons

________________________________

Charan’s comment: Rick has just published his third book:

http://www.amazon.com/Essential-Guide-Digital-Signal-Processing/dp/0133804429/ref=sr_1_1?s=books&ie=UTF8&qid=1402874933&sr=1-1

This is a great book for rank beginners and all high school students. It succeeds in explaining DSP WITHOUT equations! Got kids, siblings, you want to introduce to DSP, start here.

]]>*This blog post is by my friend Rick Lyons.*

**The Little Fruit Market
by Richard (Rick) Lyons**

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. As it turns out, what happened at that little fruit market in the 1950s has affected the lives of essentially everyone on our planet. Here’s the story.

Figure 1: The fruit market building today.

William Shockley

In 1948 the brilliant physicist William Shockley, along with John Bardeen and Walter Brattain, co-invented the transistor at Bell Laboratories in New Jersey. (Justifiably they were awarded Nobel Prizes in Physics in 1956.)

Figure 2: John Bardeen, William Shockley,

and Walter Brattain.

In 1955, deciding to move back to where he grew up, Shockley returned to California. He wanted to start his own company to commercialize semiconductor devices. Joining a college friend’s successful company, Beckman Instruments, Shockley was appointed Director of Beckman’s newly founded Shockley Semiconductor Laboratory division which bought the old fruit market at 391 San Antonio.

Shockley, who was nationally famous in the field of electronics at that time recruited the best and brightest scientists and engineers to work at his Shockley Semiconductor Laboratory. However, his domineering management style, bizarre behavior that bordered on paranoia, and his loss of interest in developing commercial transistors and integrated circuits caused eight employees to leave Shockley Labs. That was on Sept. 18, 1957, a day that was ranked by the New York Times newspaper as one of the Top 10 Days That Changed the World.

Traitorous Eight

Those “Traitorous Eight”, as Shockley called them, wanted to start their own semiconductor company. (Those engineers are shown in Figure 3. And yes, engineers really did dress that way in the late 1950s.)

Figure 3: The Traitorous Eight in 1957.

Intrigued by these new-fangled transistors, New York industrialist Sherman Fairchild agreed to finance The Traitorous Eight by creating a new company called Fairchild Camera and Instrument. It was at Fairchild that the first commercially-viable integrated circuit was invented. From that single company evolved the greatest collection of semiconductors companies in the world, as well as a California location that came to be known as “Silicon Valley.” Figure 4 gives you some idea of the astounding commercial outgrowth of Fairchild Camera and Instrument.

Figure 4: The offspring of Fairchild Camera and Instrument.

(Figure 4 is a redrawn, without permission, version of a graphic found on page 12 of the in October 2007 issue of The IEEE Spectrum magazine. A more informative version of Figure 4 is available at http://www.businessweek.com/pdfs/fairkid.pdf.)

The venture capital firm of Kleiner, Perkins, Caufield & Byers is included in Figure 4 because they provided financial backing for many electronic and information technology startup companies. You may have heard of some of those companies; Amazon, Google, Sun Microsystems, AOL, Compaq, Electronic Arts, Intuit, Netscape, and others.

Transistors, They’re Everywhere

Roughly ten years ago a technologist in the semiconductor industry estimated that mankind produces more transistors annually than grains of rice. That estimate is not as far-fetched as it might seem. In reference [2] the authors stated:

• Semiconductor production has increased by an astounding

average of 16% per year for the last forty years.

• In 2002 there were more bits of memory on a single 300

millimeter silicon wafer than were produced by the entire

semiconductor industry in 1984.

Figure 5: 300 mm silicon wafer.

• There were more transistors produced in 2002 than grains

of rice, and each rice grain could buy 100’s of transistors.

The justification for that last statement as follows: The world produced 450 billion kilograms of rice in one year. Informal measurements suggested that there are roughly 60,000 grains of rice per kilogram, meaning that approximately 27 quadrillion (27×1015) grains of rice were harvested in 2002. Assuming that one bit of semiconductor memory required at least one transistor, the semiconductor industry produced 1000 quadrillion (one quintillion, 1×1018) transistors in 2002. This amounts to about 37 transistors per grains of rice. I imagine that transistor-to-rice grain ratio was even higher in 2013.

Transistors do indeed seem to be nearly everywhere in our modern lives. I walked around the rooms of my 1500 ft2 house and realized that no matter where I stood I was never more than 3 meters from a semiconductor device. (On my bathroom counter resides a battery powered tooth brush, sitting in its recharging base.) I finally stood at the far end of my garage away from the corner where my clothes washer is located, thinking that was a spot surely more than 3 meters from a transistor. Then I realized that on the outside of the garage wall, next to where I was standing, is my electric utility company’s Smart Meter with its RF transmitter circuitry.

The inventions of the transistor and the integrated circuit (interconnected transistors) have literally transformed our world–from singing greeting cards to iPads, from home computers to interplanetary spacecraft. There is no way to overstate the importance of the technology that blossomed from that little fruit market at 391 San Antonio Road.

References

[1] http://www.businessweek.com/pdfs/fairkid.pdf.

[2] “Semiconductor Silicon – Proceedings of the Ninth International

Symposium on Silicon Materials Science and Technology”, Editors:

H. Huff, L. Fabry, and S. Kishino, 2002. See pages 125 and 142 at:

http://books.google.co.uk/books?id=cZKrzphCh-sC&pg=PA125&dq=transistors+than+grains+of+rice&hl=en&sa=X&ei=DE_wUOS5DOegiQLNroCYDA&ved=0CDYQ6AEwAA#v=onepage&q=transistors%20than%20grains%20of%20rice&f=false

[3] Interesting videos describing the rise of the semiconductor

industry can be found at:

http://en.wikipedia.org/wiki/William_Shockley

]]>*This post is by Richard Lyons, the author of the superb book “Understanding Digital Signal Processing”* – Charan Langton

_________________________

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

could cause the audio from the two engines to be multiplied

so that I can hear the low-frequency beat frequency?”

I didn’t have an answer for my friend but his question did start me thinking.

Beat Notes

You’ve probably heard of beat notes before. In Mitra’s terrific DSP book, he describes a beat note as follows [1]: If we multiply two sine wave signals, having similar frequencies, the result is a sum-frequency sine wave and a difference-frequency sine wave. Mathematically, this multiplication can be shown by a common trigonometric identity as:

The cos[2x(f1–f2)t]/2 sinusoid, the difference frequency, is called the “beat note.” If you’ve ever studied AM demodulation then you’ve seen that sum and difference Eq. (1) before.

I remember years ago when someone showed me how to tune a guitar string to the proper pitch, relative to another string, using the notion of beat notes. When you pluck two strings tuned to similar, but not identical, frequencies you can hear what seems to be a low-frequency beat note. When the two strings are tuned closer and closer in frequency the beat note becomes lower in frequency. When the two strings are tuned to the same frequency (same pitch) the beat note frequency goes to zero and can no longer be heard. In that event the two strings are properly tuned relative to each other. As was explained to me, the beat note we hear is the difference in frequency between two improperly tuned strings. What I’ve since learned is this description of a guitar’s audible beat note is NOT correct. Allow me to explain.

The Product and Sum of Two Audio Tones

Figure 1 shows two sine wave tones, an f1 = 210 Hz tone and an f2 = 200 Hz tone.

If we multiply those two Figure 1 sine waves, as indicated by Eq. (1), the product is shown as the solid curve in Figure 2. In that figure we see the solid curve is the sum-frequency (f1 + f2) 410 Hz sine wave. And the amplitude offset (the instantaneous bias) of the 410 Hz sine wave fluctuates at a rate of a 10 Hz difference frequency (f1 – f2) as shown by the red dashed curve. I’ve included the red dashed curve in Figure 2 for reference only. Again, the blue curve alone is our sin(2210t)•sin(2200t) product signal

If we were to drive a speaker with the Figure 2 product signal we’d hear the 410 Hz sum-frequency tone but we would NOT hear the 10 Hz amplitude offset. (Matlab code is provided below to demonstrate what I’m claiming here.)

As it turns out, Eq. (1) is not the expression we need when considering the sum of two sine wave tones. It’s on the following trigonometric identity that we should focus:

Equation (2), the sum of a sin(2f1t) sine wave and a sin(2f2t) sine wave, describes what happens when two guitar strings are plucked as well as what my friend hears inside a DC-9 aircraft.

If we add the two Figure 1 sine waves, as indicated by Eq. (2), the sum is shown as the solid curve in Figure 3. In that figure we see the solid curve is the sin[2(200+210)t/2] 205 Hz tone predicted by Eq. (2).

However, the peak-to-peak amplitude of that 205 Hz tone is modulated (multiplied) by a cos[2(210-200)t/2] 5 Hz sinusoid. (I’ve included the red dashed curve in Figure 3 for reference only.) Isn’t it interesting that when we add a 210 Hz sine wave to a 200 Hz sine wave the result is a fluctuating-amplitude 205 Hz sinusoidal wave? I don’t think that fact is at all intuitive. At least it wasn’t intuitive to me. That sine wave summation behavior is the “interesting observation” mentioned in the title of this blog.

In Eq. (2), with f1 = 210 and f2 = 200, I view the factor 2cos(25t) in Eq. (2) as an amplitude function that controls the amplitude of the sin(2205t) audio tone. That viewpoint is shown below.

Now if we drive a speaker with the Figure 3 sum signal we’d hear a 205 Hz tone and that tone’s amplitude (volume) would fluctuate at a rate of 10 Hz. The amplitude fluctuations occur at a 10 Hz rate because the 205 Hz tone’s amplitude goes from zero to its maximum value once for each half cycle of the 5 Hz modulating frequency. (Again, Matlab code to generate and listen to the Figure 3 signal is given below.)

Conclusion

So what this all means is that when we simultaneously pluck two guitar strings tuned to 210 Hz and 200 Hz respectively, we hear a tone whose frequency is 205 Hz (the average of 210 Hz and 200 Hz) and we also hear the 205 Hz tone’s amplitude fluctuate at a 10 Hz rate. And the oscillating amplitude fluctuations give us the impression that we hear a 10 Hz tone when in fact no 10 Hz tone exists in the Figure 3 sum signal. I say that because the spectrum of the Figure 3 solid sum-signal curve contains two spectral components, a 200 Hz tone and a 210 Hz tone. Nothing more and nothing less. As such we can say that plucking two guitar strings, off-tuned by 10 Hz, does NOT generate a sinusoidal 10 Hz audio beat note.

And to answer the DC-9 question, in my opinion no multiplication of engine noise is taking place. The audible whump whump whump sound is fluctuations in the amplitude (volume) of the two engines’ average rotational frequencies.

–Richard Lyons

Author of “Understanding Digital Signal Processing”

References

[1] S. Mitra, Digital Signal Processing, A computer-Based Approach,

McGraw-Hill, New York, New York, 2011, pp. 70-71.

Matlab Code

The following Matlab code enables you to generate, and listen to, the sum of

two audio tones.

% Filename: Beat_Frequency.m

%

% [Richard Lyons, Feb. 2013]

clear, clc

Fs = 8192; % Sample rate of dig. samples

N = 8192; % Number of time samples

n = 0:N-1;

Wave_1 = sin(2*pi*210*n/Fs); % First tone, 210 Hz

Wave_2 = sin(2*pi*200*n/Fs); % Second tone, 200 Hz

% Plot the two tones

figure(1)

plot(n/Fs, Wave_1, ‘-b’)

ylabel(‘200 Hz’); xlabel(‘Time (sec.)’);

hold on

plot(n/Fs, Wave_2, ‘-r’)

axis([0, 0.05, -1.2, 1.5]);

ylabel(‘Input tones’); xlabel(‘Time (sec.)’);

title(‘red = 200 Hz tone, blue = 210 Hz tone’);

grid on, zoom on

hold off

Product = Wave_1.*Wave_2;

Sum = Wave_1 + Wave_2;

% Plot the tones’ product and sum

figure(2)

subplot(2,1,1)

plot(n/Fs, Product, ‘-b’),

ylabel(‘Product’); xlabel(‘Time (sec.)’);

grid on, zoom on

hold on

Red_Curve = 0.5*cos(2*pi*10*n/Fs) + 0.5; % Used for plotting only

plot(n/Fs, Red_Curve, ‘-r’)

axis([0, 0.3, -1.25, 1.5]);

hold off

grid on, zoom on

subplot(2,1,2)

plot(n/Fs, Sum, ‘-b’)

hold on

Red_Curve = 2*cos(2*pi*5*n/Fs); % Used for plotting only

plot(n/Fs, Red_Curve, ‘-r’)

axis([0, 0.3, -2.4, 3]);

hold off

ylabel(‘Sum’); xlabel(‘Time (sec.)’);

grid on, zoom on

% Play all the signals

sound(Wave_1, Fs)

pause(1.2)

sound(Wave_2, Fs)

pause(1.2)

sound(Product, Fs)

pause(1.2)

sound(Sum, Fs)

% Spec analysis of the “Sum” signal

Spec = fft(Sum);

Spec_Mag = abs(Spec);

Freq = n*Fs/N; % Freq vector in Hz

figure (3) % Plot positive-freq spec amg

plot(Freq(1:N/16), Spec_Mag(1:N/16))

title(‘Spec Mag of “Sum” signal’)

ylabel(‘Mag.’), xlabel(‘Hz’)

grid on, zoom on

The site has fallen off totally from Google, probably because none of the old links can be found by Google. Hopefully their robots will find this new address.

In the meanwhile, a direct link from your site to mine will help a lot.

If you know of any good engineering tutorial sites, please also let me know.

Thanks,

Charan Langton

]]>http://complextoreal.com/tutorials/tutorial-6-3-dft-fft-part-5/#.UV0TYZOsiSo

Once you dive into the topic, you realize how simple on hand but then so complex at the same time, kind of like Life, I guess. The next topic on the list is Windows. Use of Windows is almost mandatory when doing DFT on real signals. So understanding how to use them is as important as the DFT itself.

Charan Langton

April 3, 2013

]]>

I read the first chapter that night and felt exhilarated. I had my first aha moment in DSP. Although I was out of graduate school for several years at that time, I felt that I had never really understood the subject. Yes, I could do the transforms for homework etc., but understood, not really. In this book, Lyons starts with discrete signals, goes through sampling and aliasing in the first chapter. Each chapter build gently on the previous. All just a model of clarity and beauty. I particularly loved the filter chapter, with such easy to understand exposition of what the equation meant, the forward part and the reverse part. We all love pictures and the book’s strength is its ability to communicate not just in words but also in figures. From DFT to filter design to DSP algorithms, all come alive as explained by Lyons.

I think I did read the whole book in about a week. I then flipped to the end to see who this guy was. It turned out that he worked locally at TRW. So hesitatingly, I called him to tell him how much I loved his book. He became my role model and a friend. I had been writing papers and felt that this is the way engineering should be taught. This is the way engineering books should be written. With the student in mind. No hiding behind formulas.

I recently picked up the book again as I am writing some papers on FFTs. And despite being somewhat smarter today than 15 yeas ago, I find the book still a model of engineering writing. Just a plain excellent book, deserving of all the superlatives I can muster. Fantastic, etc. etc. If you are a student in this field or an engineer, I recommend that you add this book to your library immediately.

If you have read this book, would love to hear what you think of it.

-Charan Langton

http://www.amazon.com/Understanding-Digital-Signal-Processing-3rd/dp/0137027419/ref=sr_1_1?s=books&ie=UTF8&qid=1361589879&sr=1-1&keywords=richard+lyons

]]>http://www2.egr.uh.edu/~glover/applets/Sampling/Sampling.html

Requires Java.

]]>