(So far, 65 people got it right out of 189 for a success rate of 34%)

(Thank you to Rick Lyons, author of the top-selling book Understanding Digital Signal Processing, for submitting this Quiz Question.)


This Quiz Question is related to the notion of periodicity. Think about the x(n) time sequence, containing exactly three sinusoidal cycles, defined by

$$ x = \cos (2 \pi f_0 \frac {n}{N} - 6.5 \frac {\pi}{11}) $$

shown in Figure 1(a). ($ f_0 = 3 $, $ N = 24 $ and the integer time-domain index $ n $ is $ 0 \leq n \leq N-1 $.)

We can call x(n) a "time-domain periodic" sequence meaning it has periodically repeating sample values as shown by the red and blue dots in Figure 1(b) and the x(n) sample values listed in Figure 1(d).

Sequence x(n)'s DFT spectral magnitude samples, |X(m)| having only two non-zero samples, are shown in Figure 1(c).

Quiz Question: Contemplating Figure 1, is it correct to make the following statement:

"All real-valued N-sample sinusoidal sequences containing an integer number of cycles, and having a DFT with only two non-zero spectral components, will be time-domain periodic."

Pick one:
Yes
No

[ - ]
Comment by SlartibartfastNovember 26, 2019

It still has period N in the time domain.

[ - ]
Comment by Michael CollinsNovember 27, 2019

Are you referring to the x1(n) sequence given in the answer? 

In this case, the time-domain sequences referred to here are only defined for N samples in the time domain. So, they are not "periodic" in the normal sense of the word, since any periodic signal (other than the zero signal) must be infinite in extent.

The meaning of the term "periodic" in this context is defined in the example given just above the question, as a sequence that has:

"periodically repeating sample values"

This seems to be interpreted as no sample values must repeat. Another way to phrase the question would be:

True or False:

"All real-valued N-sample sinusoidal sequences containing an integer number of cycles, and having a DFT with only two non-zero spectral components, will necessarily have repeating sample values."

SPOILER:

Another counterexample would be the 4-sample sequence:

cos(2*pi*n/4+pi/3)

n = 0,...,3.

which is given by:

0.5
-sqrt(3)/2
-0.5
sqrt(3)/2

i.e. just one cycle.

[ - ]
Comment by Rick LyonsNovember 29, 2019
Hi Michael Collins.

You are correct. The college textbook definition of a "periodic" sequence, of period N, is a sequence that satisfies:

x(n + N) = x(n), for all n.

And that restriction can only apply to infinite-length sequences! (I've been thinking about "periodicity" lately and I believe periodicity is not nearly as simple as we thought it was when we first begin our journey into the mysterious and fascinating Land of DSP!)


Your length N = 4 example, cos(2*pi*n/4+pi/3), has a period of P = 4. But to be time-domain periodic (periodically repeating sample values) its period P must less than N. So, indeed, your cos(2*pi*n/4+pi/3) is not time-domain periodic.

[ - ]
Comment by Rick LyonsNovember 30, 2019

Hi Eric. Hope you're doing well.

Please see my below Reply to Micael Collins' Comment.

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