discrete complex exponential

Hello,

My question is related to discrete signal using complex exponential for the
sake of analysis.

Assume fundamental complex exponential is exp((2*pi*)/N)*n).
Now I get harmonics of above fundamental frequency frequency such as
exp((2*pi)*/N)*nk) for different value of k.

Assume for a given value of N, if we have all the samples for the
fundamental frequency then would the samples of the harmonics be strictly
sub-set of the samples of the fundamental frequency.

I think they do and I will do some computation to understand this but
wanted a get a view here	 

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On 12/22/13 11:18 AM, Sharan123 wrote:
> Hello,
>
> My question is related to discrete signal using complex exponential for the
> sake of analysis.
>
> Assume fundamental complex exponential is exp((2*pi*)/N)*n).
> Now I get harmonics of above fundamental frequency frequency such as
> exp((2*pi)*/N)*nk) for different value of k.
>

you need a "j" (or whatever you call the imaginary unit) in there.

perhaps exp(j*(2*pi*)/N)*n)

> Assume for a given value of N, if we have all the samples for the
> fundamental frequency then would the samples of the harmonics be strictly
> sub-set of the samples of the fundamental frequency.
>
> I think they do and I will do some computation to understand this but
> wanted a get a view here	
>

this sounds a little like an NCO (numerically-controlled oscillator) or 
DDS (direct digital synthesis) or, in the computer music discipline we 
might call it additive synthesis using a sinusoidal wavetable.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Sun, 22 Dec 2013 10:18:31 -0600, Sharan123 wrote:

> Hello,
> 
> My question is related to discrete signal using complex exponential for
> the sake of analysis.
> 
> Assume fundamental complex exponential is exp((2*pi*)/N)*n).
> Now I get harmonics of above fundamental frequency frequency such as
> exp((2*pi)*/N)*nk) for different value of k.

As Robert says, you're missing i (or j) in there.  As written, that's the 
exponential that has me staring at a check plot saying "oh damn.  I did 
it again."  (You'll start by saying "what the @$%#?")

> Assume for a given value of N, if we have all the samples for the
> fundamental frequency then would the samples of the harmonics be
> strictly sub-set of the samples of the fundamental frequency.

As written, yes.  In fact, if you look at the detailed development of the 
FFT, you'll see that this repetition is what is leveraged to reduce the 
number of computations.

> I think they do and I will do some computation to understand this but
> wanted a get a view here

I hope that in your opening statement about using the complex exponential 
for analysis that you are aware that the utility of Fourier techniques 
only carries you so far; there's a lot of linear signal analysis that is 
much more profitably done using the z transform.

-- 

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

>you need a "j" (or whatever you call the imaginary unit) in there.
Right. I did miss that ...

>As written, yes.  In fact, if you look at the detailed development of the

>FFT, you'll see that this repetition is what is leveraged to reduce the 
>number of computations.

Thanks. I will look into this in details when I read fast fourier
transform.
I have not reached that far in my DSP joourney, yet.

>I hope that in your opening statement about using the complex exponential

>for analysis that you are aware that the utility of Fourier techniques 
>only carries you so far; there's a lot of linear signal analysis that is 
>much more profitably done using the z transform.

My interest is pure theory of LTI system at the moment. The part I am
reading right now deals with representing a periodic signal using complex
exponentials, analyze the output for each of these individual signal and
then generate the output for the complete input.


	 

_____________________________		
Posted through www.DSPRelated.com

On Mon, 23 Dec 2013 03:27:23 -0600, Sharan123 wrote:

>>you need a "j" (or whatever you call the imaginary unit) in there.
> Right. I did miss that ...
> 
>>As written, yes.  In fact, if you look at the detailed development of
>>the
> 
>>FFT, you'll see that this repetition is what is leveraged to reduce the
>>number of computations.
> 
> Thanks. I will look into this in details when I read fast fourier
> transform.
> I have not reached that far in my DSP joourney, yet.
> 
>>I hope that in your opening statement about using the complex
>>exponential
> 
>>for analysis that you are aware that the utility of Fourier techniques
>>only carries you so far; there's a lot of linear signal analysis that is
>>much more profitably done using the z transform.
> 
> My interest is pure theory of LTI system at the moment. The part I am
> reading right now deals with representing a periodic signal using
> complex exponentials, analyze the output for each of these individual
> signal and then generate the output for the complete input.

What book are you studying out of, or are you trying to harvest this 
stuff piecemeal off the web?

You really, really want to do this out of one good book at first -- there 
are a bunch of valid ways to do this that are each self-consistent but 
which appear to contradict each other for a surface reading.  In a book-
length treatment of this material the author should make sure that the 
material is self-consistent, even if it means going back and editing 
chapters 1 through N-1 when a contradiction is found in the last 
chapter.  When you scarf it up in pieces off the web, that won't have 
been done, and until you have a consistent understanding, it can take 
more time to try to tie everything together than it does to actually 
absorb the "good stuff".

-- 

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

Tim Wescott <tim@seemywebsite.really> writes:

> On Mon, 23 Dec 2013 03:27:23 -0600, Sharan123 wrote:
>
>>>you need a "j" (or whatever you call the imaginary unit) in there.
>> Right. I did miss that ...
>> 
>>>As written, yes.  In fact, if you look at the detailed development of
>>>the
>> 
>>>FFT, you'll see that this repetition is what is leveraged to reduce the
>>>number of computations.
>> 
>> Thanks. I will look into this in details when I read fast fourier
>> transform.
>> I have not reached that far in my DSP joourney, yet.
>> 
>>>I hope that in your opening statement about using the complex
>>>exponential
>> 
>>>for analysis that you are aware that the utility of Fourier techniques
>>>only carries you so far; there's a lot of linear signal analysis that is
>>>much more profitably done using the z transform.
>> 
>> My interest is pure theory of LTI system at the moment. The part I am
>> reading right now deals with representing a periodic signal using
>> complex exponentials, analyze the output for each of these individual
>> signal and then generate the output for the complete input.
>
> What book are you studying out of, or are you trying to harvest this 
> stuff piecemeal off the web?
>
> You really, really want to do this out of one good book at first -- there 
> are a bunch of valid ways to do this that are each self-consistent but 
> which appear to contradict each other for a surface reading.  In a book-
> length treatment of this material the author should make sure that the 
> material is self-consistent, even if it means going back and editing 
> chapters 1 through N-1 when a contradiction is found in the last 
> chapter.  When you scarf it up in pieces off the web, that won't have 
> been done, and until you have a consistent understanding, it can take 
> more time to try to tie everything together than it does to actually 
> absorb the "good stuff".

Hear hear. I was considering a similar post, but I'll just agree with
you, Tim.

Sharan, my absolute favorite book for this sort of thing is "Signals and
Systems." It's a bit dated, but linear system theory, like the laws of
physics, doesn't change.

@BOOK{signalsandsystems,
  title = "{Signals and Systems}",
  author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
  publisher = "Prentice Hall",
  year = "1983"}

-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Mon, 23 Dec 2013 14:46:22 -0500, Randy Yates wrote:

> Tim Wescott <tim@seemywebsite.really> writes:
> 
>> On Mon, 23 Dec 2013 03:27:23 -0600, Sharan123 wrote:
>>
>>>>you need a "j" (or whatever you call the imaginary unit) in there.
>>> Right. I did miss that ...
>>> 
>>>>As written, yes.  In fact, if you look at the detailed development of
>>>>the
>>> 
>>>>FFT, you'll see that this repetition is what is leveraged to reduce
>>>>the number of computations.
>>> 
>>> Thanks. I will look into this in details when I read fast fourier
>>> transform.
>>> I have not reached that far in my DSP joourney, yet.
>>> 
>>>>I hope that in your opening statement about using the complex
>>>>exponential
>>> 
>>>>for analysis that you are aware that the utility of Fourier techniques
>>>>only carries you so far; there's a lot of linear signal analysis that
>>>>is much more profitably done using the z transform.
>>> 
>>> My interest is pure theory of LTI system at the moment. The part I am
>>> reading right now deals with representing a periodic signal using
>>> complex exponentials, analyze the output for each of these individual
>>> signal and then generate the output for the complete input.
>>
>> What book are you studying out of, or are you trying to harvest this
>> stuff piecemeal off the web?
>>
>> You really, really want to do this out of one good book at first --
>> there are a bunch of valid ways to do this that are each
>> self-consistent but which appear to contradict each other for a surface
>> reading.  In a book- length treatment of this material the author
>> should make sure that the material is self-consistent, even if it means
>> going back and editing chapters 1 through N-1 when a contradiction is
>> found in the last chapter.  When you scarf it up in pieces off the web,
>> that won't have been done, and until you have a consistent
>> understanding, it can take more time to try to tie everything together
>> than it does to actually absorb the "good stuff".
> 
> Hear hear. I was considering a similar post, but I'll just agree with
> you, Tim.
> 
> Sharan, my absolute favorite book for this sort of thing is "Signals and
> Systems." It's a bit dated, but linear system theory, like the laws of
> physics, doesn't change.
> 
> @BOOK{signalsandsystems,
>   title = "{Signals and Systems}",
>   author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
>   publisher = "Prentice Hall",
>   year = "1983"}

I second Randy's book recommendation.  Not only is that book complete and 
timeless, but most of the veterans on this group (myself included) 
learned out of some variation of it, so when you ask questions your 
knowledge pool will be on the same wavelength.

-- 

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

Tim Wescott <tim@seemywebsite.really> wrote:
> On Mon, 23 Dec 2013 03:27:23 -0600, Sharan123 wrote:
 
(snip)

>> Thanks. I will look into this in details when I read fast fourier
>> transform.
>> I have not reached that far in my DSP joourney, yet.

(snip)

> What book are you studying out of, or are you trying to 
> harvest this stuff piecemeal off the web?
 
> You really, really want to do this out of one good book at 
> first -- there are a bunch of valid ways to do this that are 
> each self-consistent but which appear to contradict each other 
> for a surface reading.  

(snip)

Reminds me of the story about a Quantum Mechanics books written
using a left-handed coordinate system. All consistent, but the
minus signs will be in different places from other books.

It is supposed to be good enough, though, to keep around.

In the DSP case, that would be the signs of the exponential
in the forward and inverse Fourier transform. 

-- glen

"Sharan123" <99077@dsprelated> writes:

> Hello,
>
> My question is related to discrete signal using complex exponential for the
> sake of analysis.
>
> Assume fundamental complex exponential is exp((2*pi*)/N)*n).
> Now I get harmonics of above fundamental frequency frequency such as
> exp((2*pi)*/N)*nk) for different value of k.

Not harmonics, no. What you do have are images. (And I'm presuming
you've left out a "j" as a few others have already pointed out.)

> Assume for a given value of N, if we have all the samples for the
> fundamental frequency then would the samples of the harmonics be strictly
> sub-set of the samples of the fundamental frequency.
>
> I think they do and I will do some computation to understand this but
> wanted a get a view here	 

A view? You don't need a view, you need mathematics. 

Let's say you're sampling at Fs ==> period Ts = 1/Fs a complex
sinusoid of frequency f1. Then the baseband signal is

  exp(j * 2 * pi * f1 * n * Ts)

If -Fs/2 <= f1 < Fs/2, then there's nothing you can do to simplify
this expression. But let

  f2 = f1 + k * Fs, where k is an integer.

Then

  exp(j * 2 * pi * f2 * n * Ts) 
    = exp[j * 2 * pi * (f1 + k * Fs) * n * Ts]
    = exp[(j * 2 * pi * f1 * n * Ts) + (j * 2 * pi * k * Fs * n * Ts)]

But Fs * Ts = 1, therefore

    = exp[(j * 2 * pi * f1 * n * Ts) + (j * 2 * pi * k * n)]
    = exp[(j * 2 * pi * f1 * n * Ts)] * exp[(j * 2 * pi * k * n)]
    = exp[(j * 2 * pi * f1 * n * Ts)]
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

Randy Yates <yates@digitalsignallabs.com> writes:

> "Sharan123" <99077@dsprelated> writes:
>
>> Hello,
>>
>> My question is related to discrete signal using complex exponential for the
>> sake of analysis.
>>
>> Assume fundamental complex exponential is exp((2*pi*)/N)*n).
>> Now I get harmonics of above fundamental frequency frequency such as
>> exp((2*pi)*/N)*nk) for different value of k.
>
> Not harmonics, no. What you do have are images. (And I'm presuming
> you've left out a "j" as a few others have already pointed out.)
>
>> Assume for a given value of N, if we have all the samples for the
>> fundamental frequency then would the samples of the harmonics be strictly
>> sub-set of the samples of the fundamental frequency.
>>
>> I think they do and I will do some computation to understand this but
>> wanted a get a view here	 
>
> A view? You don't need a view, you need mathematics. 
>
> Let's say you're sampling at Fs ==> period Ts = 1/Fs a complex
> sinusoid of frequency f1. Then the baseband signal is
>
>   exp(j * 2 * pi * f1 * n * Ts)
>
> If -Fs/2 <= f1 < Fs/2, then there's nothing you can do to simplify
> this expression. But let
>
>   f2 = f1 + k * Fs, where k is an integer.
>
> Then
>
>   exp(j * 2 * pi * f2 * n * Ts) 
>     = exp[j * 2 * pi * (f1 + k * Fs) * n * Ts]
>     = exp[(j * 2 * pi * f1 * n * Ts) + (j * 2 * pi * k * Fs * n * Ts)]
>
> But Fs * Ts = 1, therefore
>
>     = exp[(j * 2 * pi * f1 * n * Ts) + (j * 2 * pi * k * n)]
>     = exp[(j * 2 * pi * f1 * n * Ts)] * exp[(j * 2 * pi * k * n)]
>     = exp[(j * 2 * pi * f1 * n * Ts)]

I should have concluded that this is one way to see why you have images
with a discrete-time complex exponential.
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com