# Upsampling problem

Started by September 21, 2008
```On Sep 24, 4:53&#2013266080;pm, zebra <e...@eastwestsounds.com> wrote:
> > you should try to design the non-linear operation or waveshaper so
> > that it inherently does not create any frequency components above fs/
> > 2. &#2013266080;at the &#2013266080;fs at which it is operating...
>
> oh right, for sure
>
> there is the always handy fact that an n-th order polynomial produces
> up to nth order harmonics. so upsample by a factor of n...
>

bingo!
```
```zebra <ezra@eastwestsounds.com> writes:

> there is the always handy fact that an n-th order polynomial produces
> up to nth order harmonics. so upsample by a factor of n...

I don't recall seeing that anywhere. Do you have a reference? Or is it
so simple it's obvious? :)
--
%  Randy Yates                  % "Watching all the days go by...
%% Fuquay-Varina, NC            %  Who are you and who am I?"
%%% 919-577-9882                % 'Mission (A World Record)',
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://www.digitalsignallabs.com
```
```Randy Yates wrote:

> zebra <ezra@eastwestsounds.com> writes:
>
>> there is the always handy fact that an n-th order polynomial produces
>> up to nth order harmonics. so upsample by a factor of n...
>
> I don't recall seeing that anywhere. Do you have a reference? Or is it
> so simple it's obvious? :)

It simply falls out in the algebra. Take a signal:

s(t) = sin (2 pi f t)

and substitute it for x in a polynomial like:

p(x) = a + b x + c x^2 + ... n x^n

Once you expand out the trig identities you will find the
maximum output component by inspection.

Erik
--
-----------------------------------------------------------------
Erik de Castro Lopo
-----------------------------------------------------------------
"It seems you are presuming a Waterfall model of development here. We're not
doing the Waterfall, we're doing the Whirlpool." -- Larry Wall on Perl6
"Perl is circling the drain, that's for sure..." --  llimllibon reddit
```
```Erik de Castro Lopo  <nospam@mega-nerd.com> wrote:

>Randy Yates wrote:

>> zebra <ezra@eastwestsounds.com> writes:

>>> there is the always handy fact that an n-th order polynomial produces
>>> up to nth order harmonics. so upsample by a factor of n...
>>
>> I don't recall seeing that anywhere. Do you have a reference? Or is it
>> so simple it's obvious? :)

>It simply falls out in the algebra. Take a signal:
>
>     s(t) = sin (2 pi f t)
>
>and substitute it for x in a polynomial like:
>
>     p(x) = a + b x + c x^2 + ... n x^n
>
>Once you expand out the trig identities you will find the
>maximum output component by inspection.

I see a distinction between the "maximum output component" of
a polynomial function being n-th order, and the polynomial function
having "up to n-th order harmonics".

Steve
```
```Erik de Castro Lopo <nospam@mega-nerd.com> writes:

> Randy Yates wrote:
>
>> zebra <ezra@eastwestsounds.com> writes:
>>
>>> there is the always handy fact that an n-th order polynomial produces
>>> up to nth order harmonics. so upsample by a factor of n...
>>
>> I don't recall seeing that anywhere. Do you have a reference? Or is it
>> so simple it's obvious? :)
>
>
> It simply falls out in the algebra. Take a signal:
>
>      s(t) = sin (2 pi f t)
>
> and substitute it for x in a polynomial like:
>
>      p(x) = a + b x + c x^2 + ... n x^n
>
> Once you expand out the trig identities you will find the
> maximum output component by inspection.

Oh yeah. Cool - thanks Erik.
--
%  Randy Yates                  % "She tells me that she likes me very much,
%% Fuquay-Varina, NC            %     but when I try to touch, she makes it
%%% 919-577-9882                %                            all too clear."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO
http://www.digitalsignallabs.com
```
```Steve Pope wrote:

> I see a distinction between the "maximum output component" of
> a polynomial function being n-th order, and the polynomial function
> having "up to n-th order harmonics".

Not "up to n-th order harmonics", "up to and including n-th order
harmonics".

Erik
--
-----------------------------------------------------------------
Erik de Castro Lopo
-----------------------------------------------------------------
"And MS thinks Linux is vulnerable to forking? 95, 95 OEM SR2, 98, 98SE,
ME, NT, 2000, Bob, .NET, CE, Datacenter, Server, Adv. Server, and now
Web Server, sheesh." -- BTS on LinuxToday.com

```