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I seek to understand symmetry properties of analytic sequences, without referring to frequency domain: what criteria must a complex sequence $x[n]$ satisfy to be analytic? Framed alternatively, such a sequence's (if also zero-mean) self-correlation (cross-correlation with self) is globally zero.

I found such conditions for $\sum_n x[n] \cdot x[n] = 0$, i.e. $\text{selfcorr[0]}=0$. Let $A$ be one complex sequence, $B$ another (in this case both are $x[n]$); then the complex product is:

$$ \begin{align} A \cdot B =\ & (A \cdot B)\text{.re } + j(A \cdot B)\text{.im } \\ =\ & [(A\text{.re } \cdot B\text{.re}) - (A\text{.im } \cdot B\text{.im})]\ + \tag{1} \\ & j[(A\text{.re } \cdot B\text{.im}) + (A\text{.im } \cdot B\text{.re})] \tag{2} \end{align} $$

  1. Real and imaginary L2 norms are equal: $\sqrt{\sum |A\text{.re}|^2} = \sqrt{\sum |A\text{.im}|^2}$, i.e. $||A\text{.re}|| = ||A\text{.im}||$. Follows from $(1)$.
  2. Real part is even-symmetric and imaginary part is odd-symmetric, or vice versa, for both of $A, B$. Follows from $(2)$ and $(1)$.

This guarantees $\sum AB = 0$, but only if $A = B$; it also guarantees $(\sum AB)\text{.im} = 0$ even if $A \neq B$ (if imag part is odd-symmetric). Note these are sufficient but not necessary for $(1)$ and $(2)$ to hold (they can be satisfied in other ways).

These aren't sufficient, however, for all $\sum_n x[n] \cdot x[n + T]$; for any general $x$, this is attained only if $x$ is analytic (or anti-analytic) and zero-mean. Self-correlation = convolution with own conjugate = freq-domain product with own conjugate: 1) if not zero mean, dc persists; 2) if both positive and negative frequencies exist, then some will persist. Thus,

  1. $x$ is zero-mean.

The most direct answer is, inner products with cisoids of opposite frequency are zero - but what's this mean, exactly, in terms of criteria on $x$ like 1, 2, and 3? (It's what I mean by "without referring to frequency domain") Note, asymmetric analytic is possible (but I'm only interested in symmetric).


Reference sequence

Code to generate $x$ with $\sum x = \sum x^2 = 0$ for reference:

import numpy as np

N = 128
x = np.random.randn(N) + 1j*np.random.randn(N)
x[N//2:] = 0

slc = x[:N//2][::-1]
x[N//2:] = slc.real - 1j * slc.imag
x -= x.mean()

x.real *= (np.sqrt(np.sum(np.abs(x.imag)**2)) /
           np.sqrt(np.sum(np.abs(x.real)**2)))

assert np.allclose(x.sum(), 0)
assert np.allclose((x*x).sum(), 0)

Visuals

Original motivation is to find inputs for which Morlet convolves to zero; visual of itself vs its complement can be helpful - also of a random sequence. Code at Github.

Morlet (analytic)

enter image description here

Morlet (analytic * anti-analytic)

enter image description here

Randn

enter image description here

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  • $\begingroup$ Interesting question. What's wrong with the usual "the imaginary part is the Hilbert transform of the real part" ? It doesn't explicitly go to the frequency domain, though perhaps that implies something in the frequency domain. $\endgroup$
    – Peter K.
    Aug 1 at 17:48
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    $\begingroup$ @PeterK. I'm unfamiliar with time-domain form of Hilbert transform, but suppose I'll find insights there - will look into. Also I'm uncertain regarding criterion 1, might be L1 norm instead, and unsure if there's such a relation for $A \neq B$; I'll revisit bit later. $\endgroup$ Aug 1 at 18:01
  • $\begingroup$ @PeterK. I seek to "qualify" $x$ for zero autocorrelation by only looking at it in time domain. Possible I'm missing something but I've not found much in Hilbert transform as aid; sure one can test x == analytic(x.real), but that doesn't reveal any properties like symmetry/norm. I suppose my actual goal is sufficient criteria for $A \neq B$. $\endgroup$ Aug 1 at 20:14
  • $\begingroup$ @OverLordGoldDragon: can you give a crisp definition of what exactly you mean by "analytic sequence". The standard definition of a continuous analytic signal is "has no negative frequencies" but since a discrete signal is periodic in frequency, that doesn't work. Related reading: andrewduncan.net/air $\endgroup$
    – Hilmar
    Aug 1 at 21:51
  • $\begingroup$ @Hilmar Interesting article. I intend "analytic" in discrete/finite sense: negative DFT bins = zero. This guarantees zero in self-convolution. $\endgroup$ Aug 1 at 22:19
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I think you may have a procedural problem here. You define "analytic sequence" as

negative DFT bins = zero.

i.e. in the frequency domain. But then you want to

I seek to understand symmetry properties of analytic sequences, without referring to frequency domain:

These two things don't go together. Typically all properties need to be derived from the definition and your definition is in the frequency domain.

If you allow this, than we can easily derive the property that you have derived:

$$ y[n] = \sum x[m]\cdot[m+n] = x[n]*x[-n]$$ where $*$ is circular convolution

In the frequency domain we get

$$ Y(k) = X(k) \cdot X(-k)$$

Now we can pop in your definition of $X(k) < 0, k < 0$ and we simply get

$$Y(k) = \begin{cases} X^2(0) & \text{ if } k= 0\\ 0 & \text{ if } k \neq 0 \end{cases}$$

which exactly is what you found. It's the square of the DC value, so it's zero if x is mean free.

If you want to maintain "not in the frequency domain", you need a time domain definition for "analytic". In continuous time, that's easy enough: there is a perfectly good time domain definition (see https://en.wikipedia.org/wiki/Hilbert_transform )

Making this discrete isn't straight forward, you probably need a circular Hilbert Transform, which is why I linked the article from Andrew Duncan

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  • $\begingroup$ "These things don't go together" disagreed. If the discrete spectrum is real-valued and analytic, the time-domain waveform is necessarily even-symmetric in real and odd-symmetric in imaginary, for example. The very basis functions build on symmetries, so I expect some equivalent time-domain description for positive-only frequency signals. I'll focus in another question. $\endgroup$ Aug 4 at 12:48
  • $\begingroup$ Regarding Hilbert, the only description I see is "imaginary part is overlap-added, $1/t$-weighted, real part" - which isn't very useful. It's possible this convolution implies something more tractable, but I don't see it. $\endgroup$ Aug 4 at 12:49
  • $\begingroup$ What I mean by "These things don't go together" is that you don't want to use the frequency domain but define "analytic" in the frequency domain. That means you ARE using the frequency domain. That's just a logical contradiction. Things get a lot easier if you just drop this constraint or maybe I misunderstand what you mean by "not using frequency domain" $\endgroup$
    – Hilmar
    Aug 4 at 13:45
  • $\begingroup$ Did you read the Wikipedia page? The time domain Hilbert integral is very well defined. It's very similar to convolving with an infinite sinc function, just a different shape. The time domain definition of Analytic signal is that real and imaginary part are Hilbert transforms of each other (plus a sign change). $\endgroup$
    – Hilmar
    Aug 4 at 13:48
  • $\begingroup$ Put differently I seek the equivalent time-domain description, or effects on time domain of the frequency domain transformation - the same way we have equivalent descriptions for time shift, conjugation, etc. Hilbert is indeed definitive but not "tractable": I can't tell the imaginary part is the Hilbert transform of real just by looking at it or applying a simple computation (unlike $e^{-j\omega t_0}X(\omega) \Leftrightarrow x(t - t_0)$). $\endgroup$ Aug 4 at 15:26

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