Signal orthogonality.

I am trying to generate orthogonal signals using the IFFT routine of dsPIC
and DAC. My setup is: -

256 point IFFT.
Sampling rate = 200KHz.
Hence, each bin resolution = 781Hz.
Frequency of interest = from 10KHz to 95KHz.

Literature says that for signals to be orthogonal, the integral of dot
product should be zero. That way, if my starting freq is 10KHz, I can have
10KHz, 15KHz, 20KHz, 25KHz, 30KHz, 35KHz, 40KHz, 45KHz, 50KHz, 55KHz,
60KHz, 65KHz, 70KHz, 75KHz, 80KHz, 85KHz, 90KHz, 95KHz = 18 sub carriers. I
checked by simulation that the dot product of these frequencies always add
up to zero. Hence these frequencies become my sub carriers.

My 2 doubts: -

1. I wanted to pack more sub carriers and wanted to know if there are more
frequencies in the band of 10KHz to 95KHz which are orthogonal.

2. When I do a FFT of the complex mix of waves, I get individual peaks of
frequency, but not the block of frequencies as described by most OFDM
literature (with a gap of 5KHz between each peak). That still makes me
believe that I can insert more sub carriers into my frequency band.

Can somebody please advice me?

Best regards.

On Dec 5, 6:41&#4294967295;am, "dsPIC_OFDM" <debrajdeb@n_o_s_p_a_m.yahoo.com>
wrote:
> I am trying to generate orthogonal signals using the IFFT routine of dsPIC
> and DAC. My setup is: -
>
> 256 point IFFT.
> Sampling rate = 200KHz.
> Hence, each bin resolution = 781Hz.
> Frequency of interest = from 10KHz to 95KHz.
>
> Literature says that for signals to be orthogonal, the integral of dot
> product should be zero. That way, if my starting freq is 10KHz, I can have
> 10KHz, 15KHz, 20KHz, 25KHz, 30KHz, 35KHz, 40KHz, 45KHz, 50KHz, 55KHz,
> 60KHz, 65KHz, 70KHz, 75KHz, 80KHz, 85KHz, 90KHz, 95KHz = 18 sub carriers. I
> checked by simulation that the dot product of these frequencies always add
> up to zero. Hence these frequencies become my sub carriers.
>
> My 2 doubts: -
>
> 1. I wanted to pack more sub carriers and wanted to know if there are more
> frequencies in the band of 10KHz to 95KHz which are orthogonal.
>
> 2. When I do a FFT of the complex mix of waves, I get individual peaks of
> frequency, but not the block of frequencies as described by most OFDM
> literature (with a gap of 5KHz between each peak). That still makes me
> believe that I can insert more sub carriers into my frequency band.
>
> Can somebody please advice me?
>
> Best regards.

Do you even understand why 10, 15, 20, etc kHz waves are orthogonal?
Understand this before looking for an answer to (1).

For (2), think about what you're asking. Is there anyway your existing
spectrum could look like a block, given a change in DFT parameters?
But why do you care about visuals? I think you lack an understanding
of the fundamentals. Read some of that literature again, or find some
more detailed literature, and revisit the DFT if you can.

Bryan

On 12/5/2010 5:41 AM, dsPIC_OFDM wrote:
> I am trying to generate orthogonal signals using the IFFT routine of dsPIC
> and DAC. My setup is: -
>
> 256 point IFFT.
> Sampling rate = 200KHz.
> Hence, each bin resolution = 781Hz.
> Frequency of interest = from 10KHz to 95KHz.
>
> Literature says that for signals to be orthogonal, the integral of dot
> product should be zero. That way, if my starting freq is 10KHz, I can have
> 10KHz, 15KHz, 20KHz, 25KHz, 30KHz, 35KHz, 40KHz, 45KHz, 50KHz, 55KHz,
> 60KHz, 65KHz, 70KHz, 75KHz, 80KHz, 85KHz, 90KHz, 95KHz = 18 sub carriers. I
> checked by simulation that the dot product of these frequencies always add
> up to zero. Hence these frequencies become my sub carriers.
>
> My 2 doubts: -
>
> 1. I wanted to pack more sub carriers and wanted to know if there are more
> frequencies in the band of 10KHz to 95KHz which are orthogonal.
>
> 2. When I do a FFT of the complex mix of waves, I get individual peaks of
> frequency, but not the block of frequencies as described by most OFDM
> literature (with a gap of 5KHz between each peak). That still makes me
> believe that I can insert more sub carriers into my frequency band.
>
> Can somebody please advice me?
>
> Best regards.

The sinusoids of the Fourier Transform are linearly independent 
(infinite time).
The sinusoids of the Fourier Series are linearly independent (periodic).
A family of sincs, spaced at their zero-crossings is linearly 
independent and can be expressed (each) as a sum of linearly independent 
sinusoids.
etc.

I don't know if that helps but .....

Fred

On Dec 5, 12:19&#4294967295;pm, Fred Marshall <fmarshall_xremove_the...@xacm.org>
wrote:
> On 12/5/2010 5:41 AM, dsPIC_OFDM wrote:
>
>
>
> > I am trying to generate orthogonal signals using the IFFT routine of dsPIC
> > and DAC. My setup is: -
>
> > 256 point IFFT.
> > Sampling rate = 200KHz.
> > Hence, each bin resolution = 781Hz.
> > Frequency of interest = from 10KHz to 95KHz.
>
> > Literature says that for signals to be orthogonal, the integral of dot
> > product should be zero. That way, if my starting freq is 10KHz, I can have
> > 10KHz, 15KHz, 20KHz, 25KHz, 30KHz, 35KHz, 40KHz, 45KHz, 50KHz, 55KHz,
> > 60KHz, 65KHz, 70KHz, 75KHz, 80KHz, 85KHz, 90KHz, 95KHz = 18 sub carriers. I
> > checked by simulation that the dot product of these frequencies always add
> > up to zero. Hence these frequencies become my sub carriers.
>
> > My 2 doubts: -
>
> > 1. I wanted to pack more sub carriers and wanted to know if there are more
> > frequencies in the band of 10KHz to 95KHz which are orthogonal.
>
> > 2. When I do a FFT of the complex mix of waves, I get individual peaks of
> > frequency, but not the block of frequencies as described by most OFDM
> > literature (with a gap of 5KHz between each peak). That still makes me
> > believe that I can insert more sub carriers into my frequency band.
>
> > Can somebody please advice me?
>
> > Best regards.
>
> The sinusoids of the Fourier Transform are linearly independent
> (infinite time).
> The sinusoids of the Fourier Series are linearly independent (periodic).
> A family of sincs, spaced at their zero-crossings is linearly
> independent and can be expressed (each) as a sum of linearly independent
> sinusoids.
> etc.
>
> I don't know if that helps but .....
>
> Fred

The sinusoids of the Fourier Series are easier to visualize because
they are spaced apart.

The Fourier transform sinusoids are tougher to comprehend because in
the FT model 1000 Hz and 1000.0000000000000000001 Hz are orthogonal to
each other because of the infinite time frame of the model. If you try
to visualize the difference between those two frequencies with real
world instruments you never could do it.  I have found FT to be a much
much tougher nut to crack than Fourier series.

On 12/05/2010 05:41 AM, dsPIC_OFDM wrote:
> I am trying to generate orthogonal signals using the IFFT routine of dsPIC
> and DAC. My setup is: -
>
> 256 point IFFT.
> Sampling rate = 200KHz.
> Hence, each bin resolution = 781Hz.
> Frequency of interest = from 10KHz to 95KHz.
>
> Literature says that for signals to be orthogonal, the integral of dot
> product should be zero. That way, if my starting freq is 10KHz, I can have
> 10KHz, 15KHz, 20KHz, 25KHz, 30KHz, 35KHz, 40KHz, 45KHz, 50KHz, 55KHz,
> 60KHz, 65KHz, 70KHz, 75KHz, 80KHz, 85KHz, 90KHz, 95KHz = 18 sub carriers. I
> checked by simulation that the dot product of these frequencies always add
> up to zero. Hence these frequencies become my sub carriers.

How did you come up with those numbers?  They're not integer multiples 
of your available bins, and that's the orthogonal signals that you have 
available to you.

Your 256 point IFFT has one _complex_ pair available at each point with 
frequency = (200kHz)/256 * n, where in ranges from 1 to 127 (0 and 128 
are special cases, and don't matter here anyway because you say you're 
not interested).  The lowest n that gets you a signal that is equal to 
or greater than 10kHz is 13.  The highest n that gets you a signal that 
is equal to or less than 95kHz is 121.  That's 109 steps, inclusive, and 
you get two channels per step, for a total that's a bit higher than 18.

> My 2 doubts: -
>
> 1. I wanted to pack more sub carriers and wanted to know if there are more
> frequencies in the band of 10KHz to 95KHz which are orthogonal.
>
> 2. When I do a FFT of the complex mix of waves, I get individual peaks of
> frequency, but not the block of frequencies as described by most OFDM
> literature (with a gap of 5KHz between each peak). That still makes me
> believe that I can insert more sub carriers into my frequency band.
>
> Can somebody please advice me?

I think you're missing the basic point of OFDM.  Perhaps you should hit 
the books until the analysis that I did for you above makes sense. 
While you're at it, you should ask yourself why, if your highest 
frequency of interest is 95kHz, and your sampling rate is 200kHz, you 
should be very worried about what's going on at 105kHz.

-- 

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

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

brent <bulegoge@columbus.rr.com> wrote:
(snip)

> The sinusoids of the Fourier Series are easier to visualize because
> they are spaced apart.
 
> The Fourier transform sinusoids are tougher to comprehend because in
> the FT model 1000 Hz and 1000.0000000000000000001 Hz are orthogonal to
> each other because of the infinite time frame of the model. If you try
> to visualize the difference between those two frequencies with real
> world instruments you never could do it.  I have found FT to be a much
> much tougher nut to crack than Fourier series.

Wow, I still remember not learning Fourier transforms when
explained by my physics TA at 8:00 AM.  I had known about the
Fourier series for years, and that did seem to make sense.
(Also, not mentioning my TA's accent, which didn't help at all.)

Just saying that the period goes to infinity wasn't enough to
make it obvious.  (Especially early in the morning, and likely
not enough sleep the night before.)   Replace the sum by an
integral (while still learning about integrals in math class).

But now it seems so obvious, that it is hard to remember 
not understanding it.

Then again, I remember a physics quiz requiring the solution
to a differential equation the day before we learned about it
in math class.  (Friction of a rope wrapped around a cylinder.)

-- glen