Hi Guys, In the past I've posted what I thought were interesting little DSP puzzles only to find that ten of you reply with the correct answer within 24 hours! So I've failed, thus far, to come up with any "puzzles" that turn out to puzzle anyone. But I'm not going to stop trying to puzzle you. OK, here goes. While not often stated in the DSP literature, when we decimate a signal sequence the spectrum of the decimated sequence experiences an amplitude loss. That makes sense because DFT amplitudes are proportional to the length of the sequence applied to the DFT. If we decimate a sequence by two, the decimated sequence is half the length of the undecimated sequence. So the DFT amplitudes of the decimated sequence will be half the DFT amplitudes of the undecimated sequence. The only place I've seen this "spectral amplitude loss from decimation" issue discussed is in Vaidyanathan's Multirate Systems book. Yesterday, I ran across a time sequence x(n) that when decimated by two, the DFT amplitudes of the decimated sequence were *equal* to the DFT amplitudes of the undecimated x(n)! Can you guess what x(n) is? See Ya', [-Rick-]
Another DSP puzzle
Started by ●December 13, 2004
Reply by ●December 13, 20042004-12-13
"Rick Lyons" <r.lyons@_BOGUS_ieee.org> wrote in message news:41bda998.87489843@news.sf.sbcglobal.net...> > > Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is? > > See Ya', > [-Rick-] >Probably not what you're after: My feeble guess: x(n) = 0, for all n
Reply by ●December 13, 20042004-12-13
r.lyons@_BOGUS_ieee.org (Rick Lyons) writes:> Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is?A sine wave at Fs/4? -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Reply by ●December 13, 20042004-12-13
Randy Yates <randy.yates@sonyericsson.com> writes:> A sine wave at Fs/4?Sorry - should be a cosine at Fs/4. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Reply by ●December 13, 20042004-12-13
Randy Yates wrote:> Randy Yates <randy.yates@sonyericsson.com> writes: > > >>A sine wave at Fs/4? > > > Sorry - should be a cosine at Fs/4.I have a problem here. If specific frequencies are immune to amplitude loss, then decimating an already bandlimited signal can't conserve the shape of a waveform. Aside from Smith's solution and assuming properly bandlimited waveforms, this seems impossible. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●December 13, 20042004-12-13
Rick Lyons wrote:> Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is? > > See Ya', > [-Rick-]Hi Rick, Normalized wavelets of the form w_jk(t) = 2^(j/2)w(2^j t - k) satisfy that property. Jitendra
Reply by ●December 13, 20042004-12-13
Rick Lyons wrote:> Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is? > > See Ya', > [-Rick-] >Seems to me that about any signal modulated at fs/4 would meet your criterion. That will create a signal where every other sample is zero. So, the decimated and pre-decimated values of the fft should be the same amplitude.
Reply by ●December 14, 20042004-12-14
Rick Lyons wrote:> Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is? > > See Ya', > [-Rick-] >Well, it couldn't be [1 0 1 0 1 0] because when you decimate that by two it's amplitude goes to zero. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●December 14, 20042004-12-14
Rick Lyons wrote:> Hi Guys, > > In the past I've posted what I thought > were interesting little DSP puzzles only to > find that ten of you reply with the > correct answer within 24 hours! So I've > failed, thus far, to come up with any > "puzzles" that turn out to puzzle anyone. > But I'm not going to stop trying to puzzle you. > > OK, here goes. While not often stated > in the DSP literature, when we decimate > a signal sequence the spectrum of the > decimated sequence experiences an > amplitude loss. That makes sense because > DFT amplitudes are proportional to the > length of the sequence applied to the > DFT. If we decimate a sequence by two, the > decimated sequence is half the length of > the undecimated sequence. So the DFT > amplitudes of the decimated sequence will be > half the DFT amplitudes of the undecimated > sequence. > > The only place I've seen this "spectral amplitude > loss from decimation" issue discussed is > in Vaidyanathan's Multirate Systems book. > > Yesterday, I ran across a time sequence x(n) > that when decimated by two, the DFT amplitudes > of the decimated sequence were *equal* to > the DFT amplitudes of the undecimated x(n)! > > Can you guess what x(n) is?My *guess* is that x(n) is such that every other sample is 0, and you decimate so as to remove the 0 samples: ...,x(-4),0,x(-2),0,x(0),0,x(2),0,x(4),... | Decimation by 2 V ...,x(-4),x(-2),x(0),x(2),x(4),... Continuing along these lines, I agree with Randy in that one (the only?) sequence that meets the criteria of the puzzle and where the decimated sequence also meets he Nyquist sampling criterion (although just barely...) is x(n) = cos(2*pi*fs/4*n) where fs is the sampling frequency before decimation. Rune
Reply by ●December 14, 20042004-12-14
Rune Allnor wrote:>Rick Lyons wrote: > > >>Hi Guys, >> >>In the past I've posted what I thought >>were interesting little DSP puzzles only to >>find that ten of you reply with the >>correct answer within 24 hours! So I've >>failed, thus far, to come up with any >>"puzzles" that turn out to puzzle anyone. >>But I'm not going to stop trying to puzzle you. >> >>OK, here goes. While not often stated >>in the DSP literature, when we decimate >>a signal sequence the spectrum of the >>decimated sequence experiences an >>amplitude loss. That makes sense because >>DFT amplitudes are proportional to the >>length of the sequence applied to the >>DFT. If we decimate a sequence by two, the >>decimated sequence is half the length of >>the undecimated sequence. So the DFT >>amplitudes of the decimated sequence will be >>half the DFT amplitudes of the undecimated >>sequence. >> >>The only place I've seen this "spectral amplitude >>loss from decimation" issue discussed is >>in Vaidyanathan's Multirate Systems book. >> >>Yesterday, I ran across a time sequence x(n) >>that when decimated by two, the DFT amplitudes >>of the decimated sequence were *equal* to >>the DFT amplitudes of the undecimated x(n)! >> >>Can you guess what x(n) is? >> >> > >My *guess* is that x(n) is such that every other sample >is 0, and you decimate so as to remove the 0 samples: > >...,x(-4),0,x(-2),0,x(0),0,x(2),0,x(4),... > >| Decimation by 2 >V > >...,x(-4),x(-2),x(0),x(2),x(4),... > >Continuing along these lines, I agree with Randy in >that one (the only?) sequence that meets the criteria >of the puzzle and where the decimated sequence also >meets he Nyquist sampling criterion (although just >barely...) is > >x(n) = cos(2*pi*fs/4*n) >where fs is the sampling frequency before decimation. > >Rune > > >Rune, I don't agree that the decimated signal even 'barely' meets the Nyquist Criterion. The problem is, after decimation the signal becomes a sequence of constant-amplitude samples, having been aliased down to D.C. This, by the way, shows that the often-used version of the Nyquist Criterion: that "f must be less than oe equal to fs," is not quite correct, and should be: " f must be less than fs" (Where f is the frequency of the highest-frequency component and fs is the sampling frequency.) As to Rick's puzzle, he does refer to the "DFT amplitudes" (plural) so he may have given away a hint there, suggesting maybe that there is more than one frequency component involved. Unfortunately, I can't think of any multiple-component signal that has zero-crossings in the right place and does not have the aliasing problem. It will be interesting to see the answer! Regards, John
