Another DSP puzzle

Jerry Avins wrote:

> >
> > -jim
> 
> You're right: I overlooked DC. After the needed low-pass filter,
> [1 0 1 0 1 0] should become [.5 .5 .5 .5 .5 .5], so is doesn't matter
> whether even samples are chosen, or odd.
> 

Yes, but the low pass filter is not "needed" here. The puzzle was about
aliasing and its effects. Consider what you,re example demonstrates: if
a signal is limited to the frequency band [0..fs/4] then we can discard
every other sample (even or odd) and the result will always have exactly
half the amplitude of the original. So for there to be any kind of
puzzle the signal needs to not be properly bandlimeted (or made
bandlimited by filtering). 
	As someone else pointed out in order for the result to have the same
amplitude as the original (twice what it would have for a properly
bandlimited signal) there must be for every frequency in the 0..fs/4
range a corresponding frequency in the fs/4..fs/2 range that gets
aliased and exactly doubles the amplitude. So the puzzle could have been
what does the above described time sequence look like.

-jim


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Hi KD,

KEDI wrote:
> Hi Rick (and Jitendra),
>
>
> I hope Jitendra doesn't mind me stepping in to clarify a few things
> here.
>

Not at all..

> >>w_jk(t) = 2^(j/2)w(2^j t - k)<<
>
> This is the standard notation in wavelet theory. However, to avoid
> confusion, I am going to replace j with n so that no one mistakes it
as
> sqrt(-1):
>
> W_nk(t) = 2^(n/2) * W(2^n * t - k)
>
> W_nk(t) is the nth "daughter wavelet" at step k. W(t) is the mother
> wavelet. (A mother wavelet can be any time-limited function such as a
> rectangle function.) This equation relates a daughter function to its
> mother. You can see a wavelet example here:
> http://mathworld.wolfram.com/HaarFunction.html
>
> 2^(n/2) is the normalizing factor to make every daughter eual in
total
> energy. nth and (n+1)th daughter wavelets differ in frequency by a
> factor of 2. Therefore, DFT of (n+1)th daugher has the same shape as
> DFT of the nth daughter decimated by 2. Nomalization will make the
DFTs
> equal in magnitude as well.
>
> This property of "2" successive daughter wavelet functions, though
> similar to what we are talking about here, is not the property we are
> looking for in "a" time sequence.
>

Thanks for the explanation. I couldn't have done better than this..

Daughter wavelets can also be viewed as filtered and resampled versions
of the mother wavelet. So one daughter wavelet can be seen as filtered
and resampled version of another daugther wavelet. A point to note here
though is that the shape need not be preserved when going from one
daughter to another.

I guess I interpreted the puzzle to be: construct a sequence that when
filtered and down-sampled will give rise to another sequence that has
the same DFT amplitudes. However, if we can't modify the shape, there
isn't much that can be done.


>
> BTW, Rick, I am a DSP newbie and I love your book! I am about to
finish
> reading every single page! Where can I get an errata sheet of your
2nd
> edition? THANK YOU!
>
>

You may find it interesting that wavelets and filter banks in DSP are
related. In fact, certain wavelets can be generated by iterating
filterbanks and the discrete wavelet transform of a sequence can be
computed using filter banks. If you have Matlab, you may want to check
an excellent wavelet toolbox (Wavelab) from Stanford University:


http://www-stat.stanford.edu/~wavelab/
Cheers
Jitendra

> Best Regards.
> 
> KD

Hi Jitendra

Thank you, for your link and info. The toolbox looks fascinating! Hope
to have some time to look at it in detail later.

Best regards.

KD

Jitendra Rayala wrote:
> Hi KD,
> You may find it interesting that wavelets and filter banks in DSP are
> related. In fact, certain wavelets can be generated by iterating
> filterbanks and the discrete wavelet transform of a sequence can be
> computed using filter banks. If you have Matlab, you may want to
check
> an excellent wavelet toolbox (Wavelab) from Stanford University:
> 
> 
> http://www-stat.stanford.edu/~wavelab/
> Cheers
> Jitendra