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A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance:

  • $y[n] = x[n] + y[n-1]$

  • $Y(z) = X(z) + Y(z) \cdot z^{-1}$

  • $H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1-z^{-1}}$

and now the complex plot of $H(z)$ gives us plenty of useful information - in this case, we have a pole at 1, corresponding to DC.

Likewise, we can look at "scale-difference" equations, where a fractional argument to the arithmetic function returns 0 by convention. Now it's the Dirichlet transform $F(z) = \sum_n (f[n] \cdot n^{-z} )$ that's helpful, keeping the variable as z for notational clarity:

  • $y[n] = x[n] + y[\frac{n}{2}]$

  • $Y(z) = X(z) + Y(z) \cdot 2^{-z}$

  • $H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1-2^{-z}}$

and now the complex plot of $H(z)$ likewise gives us plenty of useful information - in this case, we have poles at $-\frac{2 \pi i n}{\log(2)}, n \in \mathbb{Z}$.

My question - Is there some sort of transform useful for both "shift-" and "scale-"difference equations?

Here's an example of the sort of equation I'm talking about:

  • $y[n] = x[n] + y[\frac{n}{2}] + y[\frac{n-1}{2}]$

It would be great if the above could somehow, for some sort of transform, turn into something along the lines of

  • $Y(z) = X(z) + Y(z) \cdot 2^{-z} + Y(z) \cdot z^{-\frac{1}{2}} \cdot 2^{-z}$

  • $H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 - 2^{-z} - z^{-\frac{1}{2}} \cdot 2^{-z}}$

Or whatever. Some quick thinking will show that the above obviously doesn't work ("scale" and "shift" operations don't commute like that), but this illustrates what I'm getting at.

Does something like this exist?

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