What is the general formula for stacked DFT on the same signal?

Question

Let us assume a finite, aperiodic signal $x[n]$ upon which we perform DFT n times, as such:

DFT{DFT{DFT{...{DFT{$x[n]$}}...}

How can we calculate this directly instead of applying the DFT n times?

DFT is a 4th root of the identity map and a 2nd root of the (time-)reversal map. You go from here. — Jazzmaniac, Jul 18 '21 at 22:49
Does this answer your question? How do you interpret FFT of an FFT of a discrete signal? — AlexTP, Jul 18 '21 at 22:49
there are two issues. one is that DFT doesn't really give a rat's ass whether the original finite segment of signal was periodic or not. it makes it periodic by periodically extending it. $$ x[n+N] = x[n] \qquad \forall n \in \mathbb{Z} $$
The other issue is about scaling the DFT. There is a version of the DFT in which you need not worry about growth of energy every time it's applied. — robert bristow-johnson, Jul 18 '21 at 23:54
here's something that will spell out what options you have regarding scaling. look at the "unitary" scaling convention of the DFT. that might be helpful for expressing what you are doing. — robert bristow-johnson, Jul 19 '21 at 01:39

OverLordGoldDragon · Accepted Answer · 2021-07-18T23:21:41.790

1

Let $\text{DFT}_2 = \text{DFT}(\text{DFT}(...))$. Then,

$$ \begin{align} \text{DFT}_2(x[n]) &= N \cdot x[-n] \\ \text{DFT}_3(x[n]) &= N \cdot \text{DFT}(x[-n]) \\ \text{DFT}_4(x[n]) &= N^2 \cdot x[n] \end{align} $$ and thus

$$ \text{DFT}_M(x[n]) = N^{\lfloor M/2 \rfloor} \cdot \begin{cases} \sum_{n=0}^{N-1} x[n\cdot (-1)^{\lfloor M/2 \rfloor}] e^{-j2 \pi k n /N}, & M=\text{odd} \\\ x[n\cdot(-1)^{\lfloor M/2 \rfloor}], & M=\text{even} \end{cases} $$

Note that $\text{DFT}_4(x[n])/N^2 = x[n]$.

Testing

import numpy as np
from numpy.fft import fft
def dft(x):
    N = len(x)
    out = np.zeros(N, dtype='complex128')
    for k in range(N):
        for n in range(N):
            out[k] += x[n] * np.exp(-2jnp.pi  k * n / N)
    return out
def dft_M(x, M=1):
    N = len(x)
    sign = (-1)**(M//2)
if sign == -1:
    x_flip = np.zeros(N, dtype=x.dtype)
    x_flip[0] = x[0]
    x_flip[1:] = x[1:][::-1]
    x = x_flip

if M % 2 == 0:
    out = N**(M//2) * x
else:
    out = N**(M//2) * dft(x)
return out



for N in (128, 129):
    x = np.random.randn(N) + 1j*np.random.randn(N)
    assert np.allclose(fft(x), dft(x))
assert np.allclose(fft(fft(x)), dft_M(x, 2))
assert np.allclose(fft(fft(x)), dft_M(x, 2))
assert np.allclose(fft(fft(fft(x))), dft_M(x, 3))
assert np.allclose(fft(fft(fft(fft(x)))), dft_M(x, 4))

edited Jul 18 '21 at 23:21

answered Jul 18 '21 at 23:00

OverLordGoldDragon

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Since you are breaking it out to $M$ even and $M$ odd, I might express it as
$$ \mathcal{DFT}M \Big{x[n] \Big} = \begin{cases} N^{(M-1)/2} \ \sum\limits{n=0}^{N-1} x \big[n\cdot (-1)^{ (M-1)/2} \big] e^{-j2 \pi k n /N}, & M \ \text{odd} \ \ N^{M/2} \cdot x \big[n\cdot(-1)^{M/2} \big], & M \ \text{even} \ \end{cases} $$

you won't really need the $\lfloor$ floor $\rfloor$ operator. And you do really have to confirm that $x[\cdot]$ is periodic with period $N$. $$ x[n+N] = x[n] \qquad \forall n \in \mathbb{Z} $$
– robert bristow-johnson Jul 18 '21 at 23:51
@robertbristow-johnson We get $n \cdot i$ without floor, and fractional powers of $N$ (latter's valid depending on normalization). As for periodicity... sure, if we want to interpret it in terms of an infinite continuous-time function. – OverLordGoldDragon Jul 19 '21 at 01:17
if you look, there are no fractional powers of $N$ in the expression in my comment. the reason why something like periodicity needs to be explicit is because sometimes you have $x[-n]$. how are you gonna define the meaning of $x[-5]$ or similar? – robert bristow-johnson Jul 19 '21 at 01:32
1

@robertbristow-johnson Right, I missed that @ frac. On $-n$, absolutely, we define it to extend periodically - I meant on between $0$ and $N-1$, or inferring on $x(t)$ without having samples outside $0$ and $N-1$. – OverLordGoldDragon Jul 19 '21 at 01:38

What is the general formula for stacked DFT on the same signal?

1 Answers1

Testing