Hello everyone,
I am Alex and I have just started learning DSP. I was reading the book
DTSP by Oppenheim / Schafer / Buck , 2nd edition. I have a query.
Section 4.2 , Page 168 says s(t)=sigma ( -inf to inf ) [ delta ( t - nT)
]
where delta is the Dirac delta function .
Here is my query, I believe this should be the Kronecker Delta function.
The book says , x_s(t)= x_c(t)s(t)
= x_c(t)sigma ( -inf to inf ) [ delta ( t - nT) ]
= sigma ( -inf to inf ) [ x_c(nT) delta ( t - nT) ]
I believe the last step can hold only for the Kronecker delta. NOT the
Dirac delta. Could someone comment please ?
Thank you,
Alex
Basic Query from Oppenheim/ Schafer
Started by ●March 22, 2008
Reply by ●March 22, 20082008-03-22
On Sat, 22 Mar 2008 06:12:32 -0500, "Alex Miua" <radio_enthusiast_2008@yahoo.com> wrote:>Hello everyone, > >I am Alex and I have just started learning DSP. I was reading the book >DTSP by Oppenheim / Schafer / Buck , 2nd edition. I have a query. > >Section 4.2 , Page 168 says s(t)=sigma ( -inf to inf ) [ delta ( t - nT) >] > >where delta is the Dirac delta function . > >Here is my query, I believe this should be the Kronecker Delta function. > >The book says , x_s(t)= x_c(t)s(t) > = x_c(t)sigma ( -inf to inf ) [ delta ( t - nT) ] > = sigma ( -inf to inf ) [ x_c(nT) delta ( t - nT) ] > >I believe the last step can hold only for the Kronecker delta. NOT the >Dirac delta. Could someone comment please ? > >Thank you, >AlexHello Alex, Rather than page 168, you mean pages 142-143, right? I expect that your interesting question will result in lots of replies here, some of them conflicting, due to the subtle mathematical differences between a Dirac delta function and the Kronecker delta function. (Who knows, someone might even criticize me for calling them "functions".) Perhaps the following web page will help you. http://en.wikipedia.org/wiki/Kronecker_delta 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 of the continuous signal being sampled. Alex, to start learning DSP by reading Opp & Schafer's book will be difficult. That book is absolutely filled with useful DSP information, but it's presented in a *VERY* mathematically rigorous way. (As I like to say, "Opp & Schafer ain't for sissies.") Now, ... you may have: * powerful maths skills, * a pure heart, and * nerves of steel, in which case you may well be able to learn DSP from Opp & Schafer's book. Just know that there are a number of other DSP textbooks that provide a "mathematically gentler" introduction to DSP. After reading one of those books, then Opp & Schafer's book will be easier to read. Forgive me for "sticking my nose" into your business of learning DSP. Who knows? You may well be one of those rare individuals that can learn DSP, on their own, by reading Opp & Schafer. [-Rick-]
Reply by ●March 22, 20082008-03-22
Hello, indeed the equation that you mention, Alex, is (4.2), page 142. Now, the delta function used is the Dirac delta not the Kronecker delta. Besides, the Kronecker delta is not a function of the continuous time variable t. I don't think that there is a straightforward way to PROVE that x(t)d(t-to)=x(to)d(t-to), but i believe that the most clear way to see this, is as follows: The product x(t)d(t-to) is zero everywhere except at to. So the value of the product x(t)d(t-to) at to will be the product of its factors at to, namely x(to)d(0). Consequently the product x(t)d(t-to) can be writen in the equivalent expression x(to)d(t-to), in the sense that this last expression is zero everywhere except at to, as is the initial expression, and at to takes the value that also the initial expression takes at to, namely x(to)d(0). In this sense it is also true that x(t)d(t-to)=f(t)d(t-to), where x(t) and f(t) are two different functions, albeit with the property x(to)=f(to). In the interpretation of the previous paragraph i used f(t)=x(to), namely the simplest f(t) satisfying this property. Manolis
Reply by ●March 22, 20082008-03-22
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 contenttoo _large_ relative to the frequency content> of the continuous signal being sampled. > > Alex, to start learning DSP by reading Opp & Schafer's > book will be difficult. That book is absolutely > filled with useful DSP information, but it's > presented in a *VERY* mathematically rigorous way. > (As I like to say, "Opp & Schafer ain't for sissies.")Rick is too modest to point out that you should be reading his book instead. But you should be. http://tinyurl.com/2kdbkd Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●March 22, 20082008-03-22
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.
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. 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.
Thank you,
Alex
---------------------------------------------------------------------------------------------------------------------------
Alright,
coming back to the query.
After reading the responses it is clear to me why it is the Dirac
delta impulse train. When we multiply any cts function ( x(t) ) by
the Dirac delta impulse train, the values x(nT) get encoded in the
AREA around the point nT of the function ( x(t) into dirac delta
impulse train ) NOT in the height of the function ( x(t) into dirac
delta impulse train ).
So here is what is unsaid in Oppenheim is that to obtain the value
x(nT) from the function ( x(t) into dirac delta impulse train ) we
have the integrate the function from (nT - epsilon to nT + epsilon).
Which brings another point in my mind.
Define new_delta(t) ( a cts function ) = 1 at 0 and
= 0 elsewhere on the REAL
line.!! This is an extension of the Kronecker delta.
Now if I have an impulse train made of new_deltas , then we would have
the values x(nT) encoded in the HEIGHT of the product of x(t) and
impulse train of new delta.
Maybe we can use this simplification somewhere in DSP.
I look forward to hearing from you guys.
Thank you,
Alex
Reply by ●March 22, 20082008-03-22
>So here is what is unsaid in Oppenheim is that to obtain the value >x(nT) from the function ( x(t) into dirac delta impulse train ) we >have the integrate the function from (nT - epsilon to nT + epsilon).it is not mentioned because 1)it is not required to obtain the individual values x(nT) in the analysis 2)it is obvious that x(nT) would be obtained like that>Which brings another point in my mind. > >Define new_delta(t) ( a cts function ) = 1 at 0 and > = 0 elsewhere on the REAL >line.!! This is an extension of the Kronecker delta. > >Now if I have an impulse train made of new_deltas , then we would have >the values x(nT) encoded in the HEIGHT of the product of x(t) and >impulse train of new delta. > >Maybe we can use this simplification somewhere in DSP.i am afraid that your function would be a true mess because it is not the identity element of convolution, in contradiction to the Dirac delta which is. Try computing the spectrum of the modulated signal x(t)s(t) by using your function. Manolis
Reply by ●March 22, 20082008-03-22
On Mar 22, 9:56�pm, "Manolis C. Tsakiris" <el01...@mail.ntua.gr> wrote:> >So here is what is unsaid in Oppenheim is that to obtain the value > >x(nT) from the function �( x(t) into dirac delta impulse train ) �we > >have the integrate the function from (nT - epsilon to nT + epsilon). > > it is not mentioned because > 1)it is not required to obtain the individual values x(nT) in the > analysis > 2)it is obvious that x(nT) would be obtained like that > > >Which brings another point in my mind. > > >Define new_delta(t) �( �a cts function ) = 1 at 0 and > > � � � � � � � � � � � � � � � � � � � � = �0 elsewhere on the REAL > >line.!! This is an extension of the Kronecker delta. > > >Now if I have an impulse train made of new_deltas , then we would have > >the values x(nT) encoded in the HEIGHT of the product of x(t) and > >impulse train of new delta. > > >Maybe we can use this simplification somewhere in DSP. > > i am afraid that your function would be a true mess because it is not the > identity element of convolution, in contradiction to the Dirac delta which > is. Try computing the spectrum of the modulated signal x(t)s(t) by using > your function. � > > ManolisThank you for explaining that to me. -Alex.
Reply by ●March 22, 20082008-03-22
>Thank you for explaining that to me. > >-Alex. >You are welcome. Another observation about your function is that it has a zero Fourier Transform... My personal opinion is that if your math level is as high as you claim, then study from Oppenheim. However, you should strive to understand the big picture, i.e. what the text tries to say. In the paragraph that you mention the purpose is, as Rick pointed out, to find the spectrum of the modulated signal and examine when aliasing occurs. Manolis
Reply by ●March 22, 20082008-03-22
On Sat, 22 Mar 2008 06:12:32 -0500, Alex Miua wrote:> Hello everyone, > > I am Alex and I have just started learning DSP. I was reading the book > DTSP by Oppenheim / Schafer / Buck , 2nd edition. I have a query. > > Section 4.2 , Page 168 says s(t)=sigma ( -inf to inf ) [ delta ( t - nT) > ] > > where delta is the Dirac delta function . > > Here is my query, I believe this should be the Kronecker Delta function. > > The book says , x_s(t)= x_c(t)s(t) > = x_c(t)sigma ( -inf to inf ) [ delta ( t - nT) ] > = sigma ( -inf to inf ) [ x_c(nT) delta ( t - nT) > ] > > I believe the last step can hold only for the Kronecker delta. NOT the > Dirac delta. Could someone comment please ? > > Thank you, > AlexBecause x_c(t) is not dependent on n it commutes into (or out of) the summation without changing the value of the result. This is a rule very much like the similar rule for integration (i.e. if you're integrating in x then you can put a function of y inside or outside the integral, at your convenience). -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●March 23, 20082008-03-23
i guess i'll pile on. On Mar 22, 7:12 am, "Alex Miua" <radio_enthusiast_2...@yahoo.com> wrote: ...> I was reading the book > DTSP by Oppenheim / Schafer / Buck , 2nd edition. I have a query. > > Section 4.2 , Page 168 s(t)=sigma ( -inf to inf ) [ delta ( t - nT) ] > > where delta is the Dirac delta function . > > Here is my query, I believe this should be the Kronecker Delta function.it wouldn't make any sense if it were the Kronecker Delta> The book says, x_s(t)= x_c(t)s(t) > = x_c(t) sigma ( -inf to inf ) [ delta ( t - nT) ] > = sigma ( -inf to inf ) [ x_c(nT) delta ( t - nT) ] > > I believe the last step can hold only for the Kronecker delta. NOT the > Dirac delta. Could someone comment please ?x_s(t) = x_c(t) sigma ( -inf to inf ) [ delta ( t - nT) ] = sigma ( -inf to inf ) [ x_c(t) delta ( t - nT) ] = sigma ( -inf to inf ) [ x_c(nT) delta ( t - nT) ] the first step is true because x_c(t) does not depend on n. the second step is true because delta(t) = 0 for all t <> 0, so if it multiplies another function, the only value of the other function that matters is the value it takes when the argument to delta(.) is 0. 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.> Thank you,FWIW, r b-j
