Invisible wrote:
> O RLY?
> 
> My DSP textbook says the Fourier transform of the delta function yields 
> an amplitude of 1 for all frequencies. (Whereas the Fourier transform of 
> a zero signal would be a zero signal.)
> 

As has been mentioned, a function which is zero everywhere except for 
f(0)=1 is not what is meant by a delta function in this context.  The 
critical property of the delta function which your book is talking about 
is that it has an integral of 1 over any interval containing the 
`spike', whereas the function I described has an integral of zero 
everywhere.

There's another argument that can lead you to the same conclusion that 
you can't approximate any function over the reals with a Fourier 
transform, and that's to note that the cardinality of the number of 
possible Fourier representations is smaller than the cardinality of the 
number of possible functions on the reals, so there have to be some 
functions that you can't represent.

Post a reply to this message

Kevin Wampler wrote:

> There's another argument that can lead you to the same conclusion that 
> you can't approximate any function over the reals with a Fourier 
> transform, and that's to note that the cardinality of the number of 
> possible Fourier representations is smaller than the cardinality of the 
> number of possible functions on the reals, so there have to be some 
> functions that you can't represent.

Sure. For example, you might have a partial function that isn't defined 
at certain points.

None of this really interests me that much, because in the specific 
instance I'm looking at [processing digital audio and video] there 
aren't any such functions to worry about.

Post a reply to this message

Orchid XP v7 wrote:

> Sure. For example, you might have a partial function that isn't defined 
> at certain points.

Even if you restrict yourself to functions which are defined everywhere 
and are periodic with some given period the argument still applies.  For 
example, consider a really nasty function, like one which has a value of 
one at all rational points and zero everywhere else.

> None of this really interests me that much, because in the specific 
> instance I'm looking at [processing digital audio and video] there 
> aren't any such functions to worry about.

Agreed.  I hope that you find some interesting things with the audio and 
video processing stuff!  Is your goal to find some techniques which 
improve on current methods or to fiddle around with some of the current 
methods and understand them on a deeper level?

Post a reply to this message

>>> [We casually overlook the fact that JPEG doesn't work in the RGB colour 
>>> space, it works in some weird custom space to better accomodate the 
>>> peculiarities of the human eye.
>>
>>   It's not a "custom space". It's the YCbCr color space, which has been
>> used in television and video for like 50 years.
>
> OK, I rephrase: It looks pretty exotic to me. ;-)

Essentially the entire JPEG compression happens after an image
is split into grayscale channels though. The color space conversion
isn't used to increase compression.

The way I understand (wavelet?) compression, like that
in JPEG2000, is that it takes a somewhat recursive approach.
I think this part involves the wavelets... (digital wavelets being
somewhat different from analog wavelets with all the
complicated math)
The algorithm goes something like this...

1. Load the image to be compressed to memory

2. Produce an image that is the difference between
the current pixel and an average of the last few...
with a few small variations, like you can't use the pixel
above the current pixel on the top line, but you can elsewhere.
For most images this is similar to edge-detect, the resulting
image is mostly black, so highly compressible.

3. Save the difference image

4. Produce an image that is the error between the image constructed
in step 2 (when reconstructed) and the image in memory

5. replace the image in memory with the error image

6. reduce the size of the image in memory, resize,
by some fixed ratio

7. if the image in memory is larger than some size then repeat steps 2-7

8. perform some standardish compression on the saved images and
produce a file.

Post a reply to this message

> Essentially the entire JPEG compression happens after an image
> is split into grayscale channels though. The color space conversion
> isn't used to increase compression.

Yes it is.  Usually, in the new colour space, two of the channels are scaled 
by 50% as a first step that vastly decreases the file size.  You would never 
be able to get this much compression with RGB without the result looking 
much worse.

Post a reply to this message