Symmetries of analyticity / zero self-correlation

Question

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,

3. $x$ 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).

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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)

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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)

Morlet (analytic * anti-analytic)

Randn

Answer 1

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