Basic Query from Oppenheim/ Schafer

On Sat, 22 Mar 2008 11:59:40 -0400, Jerry Avins <jya@ieee.org> wrote:

>Rick Lyons wrote:
>
>   ..
>
>> But please know, all of the math rigamorole in Opp 
>> & Schafer's Section 4.2 is intended merely to 
>> introduce two concepts: [1] the periodic-in-frequency 
>> nature of a "sampled" signal, and [2] the frequency-
>> domain ambiguity, called "aliasing", that occurs 
>> in the spectrum of a sampled signal if the time between 
>> samples is too small relative to the frequency content 
>
>too _large_ relative to the frequency content
>

   (snipped by Lyons)

>Jerry

Oh shoot.  You're right Jerry.

Regards,
[-Rick-]

On Sat, 22 Mar 2008 09:26:46 -0700 (PDT), Alex Miua
<radio_enthusiast_2008@yahoo.com> wrote:

>Hi Guys!
>
>Thank you so much for your time and effort typing away at the computer
>replying to my query.
>
>Dear Rick,
>
>I 1st started reading DSP from your book : ). I read the 1st 2
>chapters of your book. The diagram on Page 39, figure 2-11, took 3-4
>FULL days of figuring out and I was very discouraged at my slow speed
>so I started reading Schafer.

Hello Alex,
   In my copy of the book, Figure 2-11 is NOT on my 
page 39.  But in any case, I wrote that material ten years 
ago when I was super interested in describing "bandpass sampling".
If I ever produce a 3rd edition to the book, I'm thinking 
that I might delete Figures 2-11, 2-12, 2-14 & 2-15.
I no longer believe that so much detail is needed by people 
trying to learn DSP for the first.     

>Also, I have an MS in Applied Math from an American University, so I
>do not think Math is a hurdle for me. But yes even then looking at a
>less math intensive presentation should be enlightening.
>
>I was read the Low Price Edition (meant for countries OUTSIDE
>america ) of BOTH the books by Schafer  / Rick Lyons. I believe the
>pages numbers are different in that compared to the edition you have.
>
>I really appreciate your advice and it is wonderful having heard from
>the great Rick Lyons. 

Oh my.  I do not deserve your words.   Most people refer 
to me as a "great pain in the neck." 

>I am planning to do DSP from your book/
>Schafer's book and Signals and Systems by Schafer. I am not in America
>so I do not have access to the wonderful amount of literature
>available in America. Please do tell me if you know better books
>though, I'll try to get my hands on them.
>
>Also long ago I had partly read" The Scientist's / Engineer's guide to
>DSP" , a famous free online book.

Alex, now that is a very good idea.  Smith knows an 
awful lot about DSP, and he has a smooth, gentle, way 
of explaining things.

Good Luck,
[-Rick-]

Rick Lyons wrote:

   ...

> Hello Alex,
>    In my copy of the book, Figure 2-11 is NOT on my 
> page 39.  But in any case, I wrote that material ten years 
> ago when I was super interested in describing "bandpass sampling".
> If I ever produce a 3rd edition to the book, I'm thinking 
> that I might delete Figures 2-11, 2-12, 2-14 & 2-15.
> I no longer believe that so much detail is needed by people 
> trying to learn DSP for the first. 

It may not be important on first reading, but it will be valued by those 
who've worked through the rest. It is the most concise and comprehensive 
account of the subject that I know how to find. Please keep it in, if 
only as an appendix.

   ...

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

robert bristow-johnson wrote:
(snip)

> btw, there are subtle issues with the Dirac delta that you won't get
> with the Kronecker delta.  if you're a pure mathematician or an
> engineer who is even more anal than me (i'm pretty anal), the Dirac
> delta "function" isn't even a true function, in the manner that
> mathematicians have defined it.  math guys *really* don't like the
> expression that you or i have above (where the Dirac delta is naked
> and unclothed by an integral).  they would say that there is no
> meaning in the mathematical expression above (that we engineers use to
> describe sampling).  i'm not that anal.

But with the usual use for it in DSP problems it soon goes
into a Fourier (integral) transform where that argument goes away.
(Similar to probability and Probability Distribution Functions.)

-- glen

On Sun, 23 Mar 2008 23:58:22 -0400, Jerry Avins <jya@ieee.org> wrote:

>Rick Lyons wrote:
>
>   ...
>
>> Hello Alex,
>>    In my copy of the book, Figure 2-11 is NOT on my 
>> page 39.  But in any case, I wrote that material ten years 
>> ago when I was super interested in describing "bandpass sampling".
>> If I ever produce a 3rd edition to the book, I'm thinking 
>> that I might delete Figures 2-11, 2-12, 2-14 & 2-15.
>> I no longer believe that so much detail is needed by people 
>> trying to learn DSP for the first. 
>
>It may not be important on first reading, but it will be valued by those 
>who've worked through the rest. It is the most concise and comprehensive 
>account of the subject that I know how to find. Please keep it in, if 
>only as an appendix.
>
>Jerry

Hi Jer,
   Thanks for the comments.

[-Rick-]

On Mar 24, 12:18 am, glen herrmannsfeldt <g...@ugcs.caltech.edu>
wrote:
> robert bristow-johnson wrote:
>
> (snip)
>
> > btw, there are subtle issues with the Dirac delta that you won't get
> > with the Kronecker delta.  if you're a pure mathematician or an
> > engineer who is even more anal than me (i'm pretty anal), the Dirac
> > delta "function" isn't even a true function, in the manner that
> > mathematicians have defined it.  math guys *really* don't like the
> > expression that you or i have above (where the Dirac delta is naked
> > and unclothed by an integral).  they would say that there is no
> > meaning in the mathematical expression above (that we engineers use to
> > describe sampling).  i'm not that anal.
>
> But with the usual use for it in DSP problems it soon goes
> into a Fourier (integral) transform where that argument goes away.
> (Similar to probability and Probability Distribution Functions.)

i agree.  if not the Fourier integral (or the integral for calculating
Fourier series coefficients), it will find it's way into the
convolution integral.  the specific equation that the math guys
declared to be meaningless (for which i emphatically disagree) is the
Fourier series representation of the Dirac comb:


    SUM{ T*delta(t - kT) } = SUM{ exp(j*2*pi*(n/T)*t) }
     k                        n

they object to me using that expression as a mathematical fact.

r b-j

robert bristow-johnson wrote:
(snip)

  the specific equation that the math guys
> declared to be meaningless (for which i emphatically disagree) is the
> Fourier series representation of the Dirac comb:

>     SUM{ T*delta(t - kT) } = SUM{ exp(j*2*pi*(n/T)*t) }
>      k                        n

> they object to me using that expression as a mathematical fact.

I was wondering about probability distribution functions, which
have the same problem.   Say you have a distribution that is part
discrete and part continuous.  You can then use delta for the discrete
part of the continuous PDF.

If you think of the sum above as a PDF, that is, the probability
of the signal having a given frequency in a continuous distribution
(frequency space) then it is delta.

The PDF is also only useful in integrals.  But them maybe mathematics
doesn't like statistics, either.

-- glen

robert bristow-johnson wrote:
> On Mar 24, 12:18 am, glen herrmannsfeldt <g...@ugcs.caltech.edu>
> wrote:
>> robert bristow-johnson wrote:
>>
>> (snip)
>>
>>> btw, there are subtle issues with the Dirac delta that you won't get
>>> with the Kronecker delta.  if you're a pure mathematician or an
>>> engineer who is even more anal than me (i'm pretty anal), the Dirac
>>> delta "function" isn't even a true function, in the manner that
>>> mathematicians have defined it.  math guys *really* don't like the
>>> expression that you or i have above (where the Dirac delta is naked
>>> and unclothed by an integral).  they would say that there is no
>>> meaning in the mathematical expression above (that we engineers use to
>>> describe sampling).  i'm not that anal.
>> But with the usual use for it in DSP problems it soon goes
>> into a Fourier (integral) transform where that argument goes away.
>> (Similar to probability and Probability Distribution Functions.)
> 
> i agree.  if not the Fourier integral (or the integral for calculating
> Fourier series coefficients), it will find it's way into the
> convolution integral.  the specific equation that the math guys
> declared to be meaningless (for which i emphatically disagree) is the
> Fourier series representation of the Dirac comb:
> 
> 
>     SUM{ T*delta(t - kT) } = SUM{ exp(j*2*pi*(n/T)*t) }
>      k                        n
> 
> they object to me using that expression as a mathematical fact.
> 
> r b-j

To make the above mathematically clean you need to choose some 
representation -- call it bob(t, x) -- that goes to delta(t) in the 
limit as x -> 0 (or infinity).  Then you can say that the _limit_ of the 
left side as x -> 0 is equal to the right side.

There really ought to be a concise way of expressing that -- perhaps 
once you accept that a dirac delta always implies integration, you can 
then accept that it also sometimes implies taking the limit of bob(t, x) 
as well.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

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