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Question about aliasing formula in Lyons DSP

Started by ldn November 6, 2014
Hi Everyone.  I am a self taught programmer who also designs hardware,  and
I am trying to get up to speed on DSP with Lyon's "Understanding DSP".

I've posted a snapshot from the book here: http://i.imgur.com/MSGVtEY.jpg

I am following everything up through equation 2-3.  "m" is any integer, so
adding 2PIm gives the same sin value.

What I don't get is how he then restricts the meaning of "m" to be an
integer multiple of "n".

This all ends up with the formula in 2-5, which makes perfect sense.

I just don't understand how it is legal to change the meaning of a variable
when rearranging an equation.  Could someone please help me make the mental
leap from equation 2-3 to equation 2-4? 

Thank you.





	 

_____________________________		
Posted through www.DSPRelated.com
"ldn" <102319@dsprelated> writes:

> Hi Everyone. I am a self taught programmer who also designs hardware, and > I am trying to get up to speed on DSP with Lyon's "Understanding DSP". > > I've posted a snapshot from the book here: http://i.imgur.com/MSGVtEY.jpg > > I am following everything up through equation 2-3. "m" is any integer, so > adding 2PIm gives the same sin value. > > What I don't get is how he then restricts the meaning of "m" to be an > integer multiple of "n". > > This all ends up with the formula in 2-5, which makes perfect sense. > > I just don't understand how it is legal to change the meaning of a variable > when rearranging an equation. Could someone please help me make the mental > leap from equation 2-3 to equation 2-4? > > Thank you.
ldn, It is legal in the sense that, if an equation is true for any integer m, then it is also true for any subset of integers m. Sure, it is less general, but that's OK. Rick's goal (I believe) is to get to equation 2-5 and show in an analytical way how you get images at k times the sample rate fs. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Randy Yates <yates@digitalsignallabs.com> writes:

> "ldn" <102319@dsprelated> writes: > >> Hi Everyone. I am a self taught programmer who also designs hardware, and >> I am trying to get up to speed on DSP with Lyon's "Understanding DSP". >> >> I've posted a snapshot from the book here: http://i.imgur.com/MSGVtEY.jpg >> >> I am following everything up through equation 2-3. "m" is any integer, so >> adding 2PIm gives the same sin value. >> >> What I don't get is how he then restricts the meaning of "m" to be an >> integer multiple of "n". >> >> This all ends up with the formula in 2-5, which makes perfect sense. >> >> I just don't understand how it is legal to change the meaning of a variable >> when rearranging an equation. Could someone please help me make the mental >> leap from equation 2-3 to equation 2-4? >> >> Thank you. > > ldn, > > It is legal in the sense that, if an equation is true for any integer m, > then it is also true for any subset of integers m. > > Sure, it is less general, but that's OK. Rick's goal (I believe) is to > get to equation 2-5 and show in an analytical way how you get images at > k times the sample rate fs.
PS: It might have been a little clarer to state after equation 2-3 something like this: "If we let m = k * n, where k is any integer, then ....", etc. -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
Randy Yates <yates@digitalsignallabs.com> writes:

> Randy Yates <yates@digitalsignallabs.com> writes: > >> "ldn" <102319@dsprelated> writes: >> >>> Hi Everyone. I am a self taught programmer who also designs hardware, and >>> I am trying to get up to speed on DSP with Lyon's "Understanding DSP". >>> >>> I've posted a snapshot from the book here: http://i.imgur.com/MSGVtEY.jpg >>> >>> I am following everything up through equation 2-3. "m" is any integer, so >>> adding 2PIm gives the same sin value. >>> >>> What I don't get is how he then restricts the meaning of "m" to be an >>> integer multiple of "n". >>> >>> This all ends up with the formula in 2-5, which makes perfect sense. >>> >>> I just don't understand how it is legal to change the meaning of a variable >>> when rearranging an equation. Could someone please help me make the mental >>> leap from equation 2-3 to equation 2-4? >>> >>> Thank you. >> >> ldn, >> >> It is legal in the sense that, if an equation is true for any integer m, >> then it is also true for any subset of integers m. >> >> Sure, it is less general, but that's OK. Rick's goal (I believe) is to >> get to equation 2-5 and show in an analytical way how you get images at >> k times the sample rate fs. > > PS: It might have been a little clarer to state after equation 2-3 > something like this: "If we let m = k * n, where k is any integer, then > ....", etc.
Doh! He DOES state that. Never mind, this PS should never have been post-scripted! -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
On Thu, 06 Nov 2014 13:22:35 -0500, Randy Yates wrote:

> "ldn" <102319@dsprelated> writes: > >> Hi Everyone. I am a self taught programmer who also designs hardware, >> and I am trying to get up to speed on DSP with Lyon's "Understanding >> DSP". >> >> I've posted a snapshot from the book here: >> http://i.imgur.com/MSGVtEY.jpg >> >> I am following everything up through equation 2-3. "m" is any integer, >> so adding 2PIm gives the same sin value. >> >> What I don't get is how he then restricts the meaning of "m" to be an >> integer multiple of "n". >> >> This all ends up with the formula in 2-5, which makes perfect sense. >> >> I just don't understand how it is legal to change the meaning of a >> variable when rearranging an equation. Could someone please help me >> make the mental leap from equation 2-3 to equation 2-4? >> >> Thank you. > > ldn, > > It is legal in the sense that, if an equation is true for any integer m, > then it is also true for any subset of integers m. > > Sure, it is less general, but that's OK. Rick's goal (I believe) is to > get to equation 2-5 and show in an analytical way how you get images at > k times the sample rate fs.
I'm pretty sure that was the goal, too. I find it kind of a rough ride getting from 2-3 to 2-5 myself, and I knew where he was trying to go. It may help to do two things: First, replace the first few words of the sentence following (2-3) with "If we further restrict m to be ...". Second, consider the middle expression in (2-3): sin(2 pi f_0 n t_s + 2 pi m) Now think how you'd go about showing, by induction, that (2-3) leads to (2-5). That is what Rick is doing here, however (as we always realize AFTER the book is in print), a few more words between (2-3) and (2-4) may have been helpful. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
>On Thu, 06 Nov 2014 13:22:35 -0500, Randy Yates wrote: > >> >> It is legal in the sense that, if an equation is true for any integer
m,
>> then it is also true for any subset of integers m. >> >> Sure, it is less general, but that's OK. Rick's goal (I believe) is to >> get to equation 2-5 and show in an analytical way how you get images at >> k times the sample rate fs. > >I'm pretty sure that was the goal, too. > >I find it kind of a rough ride getting from 2-3 to 2-5 myself, and I knew
>where he was trying to go. > >It may help to do two things: > >First, replace the first few words of the sentence following (2-3) with >"If we further restrict m to be ...". > >Second, consider the middle expression in (2-3): > >sin(2 pi f_0 n t_s + 2 pi m) > >Now think how you'd go about showing, by induction, that (2-3) leads to >(2-5). That is what Rick is doing here, however (as we always realize >AFTER the book is in print), a few more words between (2-3) and (2-4) may
>have been helpful. > >-- > >Tim Wescott >Wescott Design Services >http://www.wescottdesign.com >
Thank you guys, that explanation helped. It makes more sense to me when I read that sentence as "In those cases where m is an integer multiple of n, we can make the following leap...". ldn _____________________________ Posted through www.DSPRelated.com
"ldn" <102319@dsprelated> writes:

>>On Thu, 06 Nov 2014 13:22:35 -0500, Randy Yates wrote: >> >>> >>> It is legal in the sense that, if an equation is true for any integer > m, >>> then it is also true for any subset of integers m. >>> >>> Sure, it is less general, but that's OK. Rick's goal (I believe) is to >>> get to equation 2-5 and show in an analytical way how you get images at >>> k times the sample rate fs. >> >>I'm pretty sure that was the goal, too. >> >>I find it kind of a rough ride getting from 2-3 to 2-5 myself, and I knew > >>where he was trying to go. >> >>It may help to do two things: >> >>First, replace the first few words of the sentence following (2-3) with >>"If we further restrict m to be ...". >> >>Second, consider the middle expression in (2-3): >> >>sin(2 pi f_0 n t_s + 2 pi m) >> >>Now think how you'd go about showing, by induction, that (2-3) leads to >>(2-5). That is what Rick is doing here, however (as we always realize >>AFTER the book is in print), a few more words between (2-3) and (2-4) may > >>have been helpful. >> >>-- >> >>Tim Wescott >>Wescott Design Services >>http://www.wescottdesign.com >> > > Thank you guys, that explanation helped. > > It makes more sense to me when I read that sentence as "In those cases > where m is an integer multiple of n, we can make the following leap...". > > ldn
ldn, I think you that the notation in this presentation is not quite right. This may be a little better in that it does not violate the original definition of m: http://www.digitalsignallabs.com/aliasing.htm -- Randy Yates Digital Signal Labs http://www.digitalsignallabs.com
>"ldn" <102319@dsprelated> writes: > >>>On Thu, 06 Nov 2014 13:22:35 -0500, Randy Yates wrote: >>> >>>> >>>> It is legal in the sense that, if an equation is true for any integer >> m, >>>> then it is also true for any subset of integers m. >>>> >>>> Sure, it is less general, but that's OK. Rick's goal (I believe) is
to
>>>> get to equation 2-5 and show in an analytical way how you get images
at
>>>> k times the sample rate fs. >>> >>>I'm pretty sure that was the goal, too. >>> >>>I find it kind of a rough ride getting from 2-3 to 2-5 myself, and I
knew
>> >>>where he was trying to go. >>> >>>It may help to do two things: >>> >>>First, replace the first few words of the sentence following (2-3) with
>>>"If we further restrict m to be ...". >>> >>>Second, consider the middle expression in (2-3): >>> >>>sin(2 pi f_0 n t_s + 2 pi m) >>> >>>Now think how you'd go about showing, by induction, that (2-3) leads to
>>>(2-5). That is what Rick is doing here, however (as we always realize >>>AFTER the book is in print), a few more words between (2-3) and (2-4)
may
>> >>>have been helpful. >>> >>>-- >>> >>>Tim Wescott >>>Wescott Design Services >>>http://www.wescottdesign.com >>> >> >> Thank you guys, that explanation helped. >> >> It makes more sense to me when I read that sentence as "In those cases >> where m is an integer multiple of n, we can make the following
leap...".
>> >> ldn > >ldn, > >I think you that the notation in this presentation is not quite right. >This may be a little better in that it does not violate the original >definition of m: > > http://www.digitalsignallabs.com/aliasing.htm > >-- >Randy Yates >Digital Signal Labs >http://www.digitalsignallabs.com >
Randy, Even clearer, thank you. ldn _____________________________ Posted through www.DSPRelated.com
On Thu, 06 Nov 2014 11:56:50 -0600, "ldn" <102319@dsprelated> wrote:

>Hi Everyone. I am a self taught programmer who also designs hardware, and >I am trying to get up to speed on DSP with Lyon's "Understanding DSP". > >I've posted a snapshot from the book here: http://i.imgur.com/MSGVtEY.jpg > >I am following everything up through equation 2-3. "m" is any integer, so >adding 2PIm gives the same sin value. > >What I don't get is how he then restricts the meaning of "m" to be an >integer multiple of "n". > >This all ends up with the formula in 2-5, which makes perfect sense. > >I just don't understand how it is legal to change the meaning of a variable >when rearranging an equation. Could someone please help me make the mental >leap from equation 2-3 to equation 2-4? > >Thank you.
Hello ldn, I just saw your original post today. I concur with Randy's and Tim's comments here. I apologize if my explanation of how to go from Eq. (2-3) to Eq. (2-5) was a bit difficult to understand. (I'll try to do better next time.) ldn, I've never seen my book's material in the form as shown in http://i.imgur.com/MSGVtEY.jpg before (with the equation numbers above the equations). I'm curious, ...from where did you obtain that image? If you send me a private e-mail I'll send you the errata to the 3rd edition of my book. [-Rick Lyons-]