Posit AI Weblog: Discrete Fourier Remodel

Notice: This put up is an excerpt from the forthcoming e-book, Deep Studying and Scientific Computing with R torch. The chapter in query is on the Discrete Fourier Remodel (DFT), and is situated partially three. Half three is devoted to scientific computation past deep studying.
There are two chapters on the Fourier Remodel. The primary strives to, in as “verbal” and lucid a approach as was attainable to me, solid a light-weight on what’s behind the magic; it additionally exhibits how, surprisingly, you possibly can code the DFT in merely half a dozen strains. The second focuses on quick implementation (the Quick Fourier Remodel, or FFT), once more with each conceptual/explanatory in addition to sensible, code-it-yourself components.
Collectively, these cowl way more materials than might sensibly match right into a weblog put up; due to this fact, please take into account what follows extra as a “teaser” than a completely fledged article.

Within the sciences, the Fourier Remodel is nearly in every single place. Acknowledged very typically, it converts information from one illustration to a different, with none lack of data (if finished appropriately, that’s.) In case you use torch, it’s only a operate name away: torch_fft_fft() goes a technique, torch_fft_ifft() the opposite. For the consumer, that’s handy – you “simply” have to know interpret the outcomes. Right here, I need to assist with that. We begin with an instance operate name, enjoying round with its output, after which, attempt to get a grip on what’s going on behind the scenes.

Understanding the output of torch_fft_fft()

As we care about precise understanding, we begin from the only attainable instance sign, a pure cosine that performs one revolution over the entire sampling interval.

Place to begin: A cosine of frequency 1

The way in which we set issues up, there will likely be sixty-four samples; the sampling interval thus equals N = 64. The content material of frequency(), the beneath helper operate used to assemble the sign, displays how we symbolize the cosine. Particularly:

[
f(x) = cos(frac{2 pi}{N} k x)
]

Right here (x) values progress over time (or house), and (ok) is the frequency index. A cosine is periodic with interval (2 pi); so if we would like it to first return to its beginning state after sixty-four samples, and (x) runs between zero and sixty-three, we’ll need (ok) to be equal to (1). Like that, we’ll attain the preliminary state once more at place (x = frac{2 pi}{64} * 1 * 64).

Let’s shortly verify this did what it was speculated to:

df <- information.body(x = sample_positions, y = as.numeric(x))

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("amplitude") +
  theme_minimal()

Now that we’ve the enter sign, torch_fft_fft() computes for us the Fourier coefficients, that’s, the significance of the varied frequencies current within the sign. The variety of frequencies thought of will equal the variety of sampling factors: So (X) will likely be of size sixty-four as effectively.

(In our instance, you’ll discover that the second half of coefficients will equal the primary in magnitude. That is the case for each real-valued sign. In such circumstances, you may name torch_fft_rfft() as a substitute, which yields “nicer” (within the sense of shorter) vectors to work with. Right here although, I need to clarify the final case, since that’s what you’ll discover finished in most expositions on the subject.)

Even with the sign being actual, the Fourier coefficients are complicated numbers. There are 4 methods to examine them. The primary is to extract the true half:

[1]  0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
[57] 0 0 0 0 0 0 0 32

Solely a single coefficient is non-zero, the one at place 1. (We begin counting from zero, and should discard the second half, as defined above.)

Now trying on the imaginary half, we discover it’s zero all through:

[1]  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[57] 0 0 0 0 0 0 0 0

At this level we all know that there’s only a single frequency current within the sign, specifically, that at (ok = 1). This matches (and it higher needed to) the way in which we constructed the sign: specifically, as undertaking a single revolution over the entire sampling interval.

Since, in concept, each coefficient might have non-zero actual and imaginary components, typically what you’d report is the magnitude (the sq. root of the sum of squared actual and imaginary components):

[1]  0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
[57] 0 0 0 0 0 0 0 32

Unsurprisingly, these values precisely mirror the respective actual components.

Lastly, there’s the part, indicating a attainable shift of the sign (a pure cosine is unshifted). In torch, we’ve torch_angle() complementing torch_abs(), however we have to have in mind roundoff error right here. We all know that in every however a single case, the true and imaginary components are each precisely zero; however as a consequence of finite precision in how numbers are offered in a pc, the precise values will typically not be zero. As a substitute, they’ll be very small. If we take one among these “pretend non-zeroes” and divide it by one other, as occurs within the angle calculation, large values may result. To forestall this from taking place, our customized implementation rounds each inputs earlier than triggering the division.

part <- operate(Ft, threshold = 1e5) {
  torch_atan2(
    torch_abs(torch_round(Ft$imag * threshold)),
    torch_abs(torch_round(Ft$actual * threshold))
  )
}

as.numeric(part(Ft)) %>% spherical(5)

[1]  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[57] 0 0 0 0 0 0 0 0

As anticipated, there is no such thing as a part shift within the sign.

Let’s visualize what we discovered.

create_plot <- operate(x, y, amount) {
  df <- information.body(
    x_ = x,
    y_ = as.numeric(y) %>% spherical(5)
  )
  ggplot(df, aes(x = x_, y = y_)) +
    geom_col() +
    xlab("frequency") +
    ylab(amount) +
    theme_minimal()
}

p_real <- create_plot(
  sample_positions,
  real_part,
  "actual half"
)
p_imag <- create_plot(
  sample_positions,
  imag_part,
  "imaginary half"
)
p_magnitude <- create_plot(
  sample_positions,
  magnitude,
  "magnitude"
)
p_phase <- create_plot(
  sample_positions,
  part(Ft),
  "part"
)

p_real + p_imag + p_magnitude + p_phase

It’s honest to say that we’ve no purpose to doubt what torch_fft_fft() has finished. However with a pure sinusoid like this, we will perceive precisely what’s happening by computing the DFT ourselves, by hand. Doing this now will considerably assist us later, once we’re writing the code.

Reconstructing the magic

One caveat about this part. With a subject as wealthy because the Fourier Remodel, and an viewers who I think about to differ broadly on a dimension of math and sciences schooling, my probabilities to fulfill your expectations, expensive reader, have to be very near zero. Nonetheless, I need to take the chance. In case you’re an skilled on this stuff, you’ll anyway be simply scanning the textual content, searching for items of torch code. In case you’re reasonably conversant in the DFT, you should still like being reminded of its interior workings. And – most significantly – in the event you’re reasonably new, and even utterly new, to this subject, you’ll hopefully take away (not less than) one factor: that what looks as if one of many best wonders of the universe (assuming there’s a actuality in some way similar to what goes on in our minds) could be a marvel, however neither “magic” nor a factor reserved to the initiated.

In a nutshell, the Fourier Remodel is a foundation transformation. Within the case of the DFT – the Discrete Fourier Remodel, the place time and frequency representations each are finite vectors, not capabilities – the brand new foundation appears to be like like this:

[
begin{aligned}
&mathbf{w}^{0n}_N = e^{ifrac{2 pi}{N}* 0 * n} = 1
&mathbf{w}^{1n}_N = e^{ifrac{2 pi}{N}* 1 * n} = e^{ifrac{2 pi}{N} n}
&mathbf{w}^{2n}_N = e^{ifrac{2 pi}{N}* 2 * n} = e^{ifrac{2 pi}{N}2n}& …
&mathbf{w}^{(N-1)n}_N = e^{ifrac{2 pi}{N}* (N-1) * n} = e^{ifrac{2 pi}{N}(N-1)n}
end{aligned}
]

Right here (N), as earlier than, is the variety of samples (64, in our case); thus, there are (N) foundation vectors. With (ok) operating via the premise vectors, they are often written:

[
mathbf{w}^{kn}_N = e^{ifrac{2 pi}{N}k n}
] {#eq-dft-1}

Like (ok), (n) runs from (0) to (N-1). To grasp what these foundation vectors are doing, it’s useful to briefly change to a shorter sampling interval, (N = 4), say. If we achieve this, we’ve 4 foundation vectors: (mathbf{w}^{0n}_N), (mathbf{w}^{1n}_N), (mathbf{w}^{2n}_N), and (mathbf{w}^{3n}_N). The primary one appears to be like like this:

[
mathbf{w}^{0n}_N
=
begin{bmatrix}
e^{ifrac{2 pi}{4}* 0 * 0}
e^{ifrac{2 pi}{4}* 0 * 1}
e^{ifrac{2 pi}{4}* 0 * 2}
e^{ifrac{2 pi}{4}* 0 * 3}
end{bmatrix}
=
begin{bmatrix}
1
1
1
1
end{bmatrix}
]

The second, like so:

[
mathbf{w}^{1n}_N
=
begin{bmatrix}
e^{ifrac{2 pi}{4}* 1 * 0}
e^{ifrac{2 pi}{4}* 1 * 1}
e^{ifrac{2 pi}{4}* 1 * 2}
e^{ifrac{2 pi}{4}* 1 * 3}
end{bmatrix}
=
begin{bmatrix}
1
e^{ifrac{pi}{2}}
e^{i pi}
e^{ifrac{3 pi}{4}}
end{bmatrix}
=
begin{bmatrix}
1
i
-1
-i
end{bmatrix}
]

That is the third:

[
mathbf{w}^{2n}_N
=
begin{bmatrix}
e^{ifrac{2 pi}{4}* 2 * 0}
e^{ifrac{2 pi}{4}* 2 * 1}
e^{ifrac{2 pi}{4}* 2 * 2}
e^{ifrac{2 pi}{4}* 2 * 3}
end{bmatrix}
=
begin{bmatrix}
1
e^{ipi}
e^{i 2 pi}
e^{ifrac{3 pi}{2}}
end{bmatrix}
=
begin{bmatrix}
1
-1
1
-1
end{bmatrix}
]

And at last, the fourth:

[
mathbf{w}^{3n}_N
=
begin{bmatrix}
e^{ifrac{2 pi}{4}* 3 * 0}
e^{ifrac{2 pi}{4}* 3 * 1}
e^{ifrac{2 pi}{4}* 3 * 2}
e^{ifrac{2 pi}{4}* 3 * 3}
end{bmatrix}
=
begin{bmatrix}
1
e^{ifrac{3 pi}{2}}
e^{i 3 pi}
e^{ifrac{9 pi}{2}}
end{bmatrix}
=
begin{bmatrix}
1
-i
-1
i
end{bmatrix}
]

We are able to characterize these 4 foundation vectors when it comes to their “pace”: how briskly they transfer across the unit circle. To do that, we merely take a look at the rightmost column vectors, the place the ultimate calculation outcomes seem. The values in that column correspond to positions pointed to by the revolving foundation vector at completely different closing dates. Which means that a single “replace of place”, we will see how briskly the vector is transferring in a single time step.

Wanting first at (mathbf{w}^{0n}_N), we see that it doesn’t transfer in any respect. (mathbf{w}^{1n}_N) goes from (1) to (i) to (-1) to (-i); yet another step, and it might be again the place it began. That’s one revolution in 4 steps, or a step measurement of (frac{pi}{2}). Then (mathbf{w}^{2n}_N) goes at double that tempo, transferring a distance of (pi) alongside the circle. That approach, it finally ends up finishing two revolutions general. Lastly, (mathbf{w}^{3n}_N) achieves three full loops, for a step measurement of (frac{3 pi}{2}).

The factor that makes these foundation vectors so helpful is that they’re mutually orthogonal. That’s, their dot product is zero:

[
langle mathbf{w}^{kn}_N, mathbf{w}^{ln}_N rangle = sum_{n=0}^{N-1} ({e^{ifrac{2 pi}{N}k n}})^* e^{ifrac{2 pi}{N}l n} = sum_{n=0}^{N-1} ({e^{-ifrac{2 pi}{N}k n}})e^{ifrac{2 pi}{N}l n} = 0
] {#eq-dft-2}

Let’s take, for instance, (mathbf{w}^{2n}_N) and (mathbf{w}^{3n}_N). Certainly, their dot product evaluates to zero.

[
begin{bmatrix}
1 & -1 & 1 & -1
end{bmatrix}
begin{bmatrix}
1
-i
-1
i
end{bmatrix}
=
1 + i + (-1) + (-i) = 0
]

Now, we’re about to see how the orthogonality of the Fourier foundation considerably simplifies the calculation of the DFT. Did you discover the similarity between these foundation vectors and the way in which we wrote the instance sign? Right here it’s once more:

[
f(x) = cos(frac{2 pi}{N} k x)
]

If we handle to symbolize this operate when it comes to the premise vectors (mathbf{w}^{kn}_N = e^{ifrac{2 pi}{N}ok n}), the interior product between the operate and every foundation vector will likely be both zero (the “default”) or a a number of of 1 (in case the operate has a part matching the premise vector in query). Fortunately, sines and cosines can simply be transformed into complicated exponentials. In our instance, that is how that goes:

[
begin{aligned}
mathbf{x}_n &= cos(frac{2 pi}{64} n)
&= frac{1}{2} (e^{ifrac{2 pi}{64} n} + e^{-ifrac{2 pi}{64} n})
&= frac{1}{2} (e^{ifrac{2 pi}{64} n} + e^{ifrac{2 pi}{64} 63n})
&= frac{1}{2} (mathbf{w}^{1n}_N + mathbf{w}^{63n}_N)
end{aligned}
]

Right here step one instantly outcomes from Euler’s formulation, and the second displays the truth that the Fourier coefficients are periodic, with frequency -1 being the identical as 63, -2 equaling 62, and so forth.

Now, the (ok)th Fourier coefficient is obtained by projecting the sign onto foundation vector (ok).

As a result of orthogonality of the premise vectors, solely two coefficients is not going to be zero: these for (mathbf{w}^{1n}_N) and (mathbf{w}^{63n}_N). They’re obtained by computing the interior product between the operate and the premise vector in query, that’s, by summing over (n). For every (n) ranging between (0) and (N-1), we’ve a contribution of (frac{1}{2}), leaving us with a ultimate sum of (32) for each coefficients. For instance, for (mathbf{w}^{1n}_N):

[
begin{aligned}
X_1 &= langle mathbf{w}^{1n}_N, mathbf{x}_n rangle
&= langle mathbf{w}^{1n}_N, frac{1}{2} (mathbf{w}^{1n}_N + mathbf{w}^{63n}_N) rangle
&= frac{1}{2} * 64
&= 32
end{aligned}
]

And analogously for (X_{63}).

Now, trying again at what torch_fft_fft() gave us, we see we had been capable of arrive on the similar end result. And we’ve realized one thing alongside the way in which.

So long as we stick with alerts composed of a number of foundation vectors, we will compute the DFT on this approach. On the finish of the chapter, we’ll develop code that may work for all alerts, however first, let’s see if we will dive even deeper into the workings of the DFT. Three issues we’ll need to discover:

• What would occur if frequencies modified – say, a melody had been sung at a better pitch?

• What about amplitude modifications – say, the music had been performed twice as loud?

• What about part – e.g., there have been an offset earlier than the piece began?

In all circumstances, we’ll name torch_fft_fft() solely as soon as we’ve decided the end result ourselves.

And at last, we’ll see how complicated sinusoids, made up of various elements, can nonetheless be analyzed on this approach, offered they are often expressed when it comes to the frequencies that make up the premise.

Various frequency

Assume we quadrupled the frequency, giving us a sign that seemed like this:

[
mathbf{x}_n = cos(frac{2 pi}{N}*4*n)
]

Following the identical logic as above, we will specific it like so:

[
mathbf{x}_n = frac{1}{2} (mathbf{w}^{4n}_N + mathbf{w}^{60n}_N)
]

We already see that non-zero coefficients will likely be obtained just for frequency indices (4) and (60). Choosing the previous, we receive

[
begin{aligned}
X_4 &= langle mathbf{w}^{4n}_N, mathbf{x}_n rangle
&= langle mathbf{w}^{4n}_N, frac{1}{2} (mathbf{w}^{4n}_N + mathbf{w}^{60n}_N) rangle
&= 32
end{aligned}
]

For the latter, we’d arrive on the similar end result.

Now, let’s be sure our evaluation is appropriate. The next code snippet comprises nothing new; it generates the sign, calculates the DFT, and plots them each.

x <- torch_cos(frequency(4, N) * sample_positions)

plot_ft <- operate(x)  plot_spacer()) /
    (p_real 

plot_ft(x)

This does certainly verify our calculations.

A particular case arises when sign frequency rises to the best one “allowed”, within the sense of being detectable with out aliasing. That would be the case at one half of the variety of sampling factors. Then, the sign will appear to be so:

[
mathbf{x}_n = frac{1}{2} (mathbf{w}^{32n}_N + mathbf{w}^{32n}_N)
]

Consequently, we find yourself with a single coefficient, similar to a frequency of 32 revolutions per pattern interval, of double the magnitude (64, thus). Listed below are the sign and its DFT:

x <- torch_cos(frequency(32, N) * sample_positions)
plot_ft(x)

Various amplitude

Now, let’s take into consideration what occurs once we differ amplitude. For instance, say the sign will get twice as loud. Now, there will likely be a multiplier of two that may be taken outdoors the interior product. In consequence, the one factor that modifications is the magnitude of the coefficients.

Let’s confirm this. The modification is predicated on the instance we had earlier than the final one, with 4 revolutions over the sampling interval:

x <- 2 * torch_cos(frequency(4, N) * sample_positions)
plot_ft(x)

To date, we’ve not as soon as seen a coefficient with non-zero imaginary half. To alter this, we add in part.

Including part

Altering the part of a sign means shifting it in time. Our instance sign is a cosine, a operate whose worth is 1 at (t=0). (That additionally was the – arbitrarily chosen – place to begin of the sign.)

Now assume we shift the sign ahead by (frac{pi}{2}). Then the height we had been seeing at zero strikes over to (frac{pi}{2}); and if we nonetheless begin “recording” at zero, we should discover a worth of zero there. An equation describing that is the next. For comfort, we assume a sampling interval of (2 pi) and (ok=1), in order that the instance is an easy cosine:

[
f(x) = cos(x – phi)
]

The minus signal might look unintuitive at first. But it surely does make sense: We now need to receive a price of 1 at (x=frac{pi}{2}), so (x – phi) ought to consider to zero. (Or to any a number of of (pi).) Summing up, a delay in time will seem as a unfavourable part shift.

Now, we’re going to calculate the DFT for a shifted model of our instance sign. However in the event you like, take a peek on the phase-shifted model of the time-domain image now already. You’ll see {that a} cosine, delayed by (frac{pi}{2}), is nothing else than a sine beginning at 0.

To compute the DFT, we observe our familiar-by-now technique. The sign now appears to be like like this:

[
mathbf{x}_n = cos(frac{2 pi}{N}*4*x – frac{pi}{2})
]

First, we specific it when it comes to foundation vectors:

[
begin{aligned}
mathbf{x}_n &= cos(frac{2 pi}{64} 4 n – frac{pi}{2})
&= frac{1}{2} (e^{ifrac{2 pi}{64} 4n – frac{pi}{2}} + e^{ifrac{2 pi}{64} 60n – frac{pi}{2}})
&= frac{1}{2} (e^{ifrac{2 pi}{64} 4n} e^{-i frac{pi}{2}} + e^{ifrac{2 pi}{64} 60n} e^{ifrac{pi}{2}})
&= frac{1}{2} (e^{-i frac{pi}{2}} mathbf{w}^{4n}_N + e^{i frac{pi}{2}} mathbf{w}^{60n}_N)
end{aligned}
]

Once more, we’ve non-zero coefficients just for frequencies (4) and (60). However they’re complicated now, and each coefficients are now not an identical. As a substitute, one is the complicated conjugate of the opposite. First, (X_4):

[
begin{aligned}
X_4 &= langle mathbf{w}^{4n}_N, mathbf{x}_n rangle
&=langle mathbf{w}^{4n}_N, frac{1}{2} (e^{-i frac{pi}{2}} mathbf{w}^{4n}_N + e^{i frac{pi}{2}} mathbf{w}^{60n}_N) rangle
&= 32 *e^{-i frac{pi}{2}}
&= -32i
end{aligned}
]

And right here, (X_{60}):

[
begin{aligned}
X_{60} &= langle mathbf{w}^{60n}_N, mathbf{x}_N rangle
&= 32 *e^{i frac{pi}{2}}
&= 32i
end{aligned}
]

As common, we examine our calculation utilizing torch_fft_fft().

x <- torch_cos(frequency(4, N) * sample_positions - pi / 2)

plot_ft(x)

For a pure sine wave, the non-zero Fourier coefficients are imaginary. The part shift within the coefficients, reported as (frac{pi}{2}), displays the time delay we utilized to the sign.

Lastly – earlier than we write some code – let’s put all of it collectively, and take a look at a wave that has greater than a single sinusoidal part.

Superposition of sinusoids

The sign we assemble should be expressed when it comes to the premise vectors, however it’s now not a pure sinusoid. As a substitute, it’s a linear mixture of such:

[
begin{aligned}
mathbf{x}_n &= 3 sin(frac{2 pi}{64} 4n) + 6 cos(frac{2 pi}{64} 2n) +2cos(frac{2 pi}{64} 8n)
end{aligned}
]

I received’t undergo the calculation intimately, however it’s no completely different from the earlier ones. You compute the DFT for every of the three elements, and assemble the outcomes. With none calculation, nonetheless, there’s fairly just a few issues we will say:

• Because the sign consists of two pure cosines and one pure sine, there will likely be 4 coefficients with non-zero actual components, and two with non-zero imaginary components. The latter will likely be complicated conjugates of one another.
• From the way in which the sign is written, it’s simple to find the respective frequencies, as effectively: The all-real coefficients will correspond to frequency indices 2, 8, 56, and 62; the all-imaginary ones to indices 4 and 60.
• Lastly, amplitudes will end result from multiplying with (frac{64}{2}) the scaling elements obtained for the person sinusoids.

Let’s examine:

x <- 3 * torch_sin(frequency(4, N) * sample_positions) +
  6 * torch_cos(frequency(2, N) * sample_positions) +
  2 * torch_cos(frequency(8, N) * sample_positions)

plot_ft(x)

Now, how will we calculate the DFT for much less handy alerts?

Coding the DFT

Fortuitously, we already know what must be finished. We need to venture the sign onto every of the premise vectors. In different phrases, we’ll be computing a bunch of interior merchandise. Logic-wise, nothing modifications: The one distinction is that typically, it is not going to be attainable to symbolize the sign when it comes to just some foundation vectors, like we did earlier than. Thus, all projections will truly need to be calculated. However isn’t automation of tedious duties one factor we’ve computer systems for?

Let’s begin by stating enter, output, and central logic of the algorithm to be carried out. As all through this chapter, we keep in a single dimension. The enter, thus, is a one-dimensional tensor, encoding a sign. The output is a one-dimensional vector of Fourier coefficients, of the identical size because the enter, every holding details about a frequency. The central thought is: To acquire a coefficient, venture the sign onto the corresponding foundation vector.

To implement that concept, we have to create the premise vectors, and for each, compute its interior product with the sign. This may be finished in a loop. Surprisingly little code is required to perform the aim:

dft <- operate(x) {
  n_samples <- size(x)

  n <- torch_arange(0, n_samples - 1)$unsqueeze(1)

  Ft <- torch_complex(
    torch_zeros(n_samples), torch_zeros(n_samples)
  )

  for (ok in 0:(n_samples - 1)) {
    w_k <- torch_exp(-1i * 2 * pi / n_samples * ok * n)
    dot <- torch_matmul(w_k, x$to(dtype = torch_cfloat()))
    Ft[k + 1] <- dot
  }
  Ft
}

To check the implementation, we will take the final sign we analysed, and evaluate with the output of torch_fft_fft().

[1]  0 0 192 0 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
[57] 64 0 0 0 0 0 192 0

[1]  0 0 0 0 -96 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[29] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
[57] 0 0 0 0 96 0 0 0

Reassuringly – in the event you look again – the outcomes are the identical.

Above, did I say “little code”? The truth is, a loop is just not even wanted. As a substitute of working with the premise vectors one-by-one, we will stack them in a matrix. Then every row will maintain the conjugate of a foundation vector, and there will likely be (N) of them. The columns correspond to positions (0) to (N-1); there will likely be (N) of them as effectively. For instance, that is how the matrix would search for (N=4):

[
mathbf{W}_4
=
begin{bmatrix}
e^{-ifrac{2 pi}{4}* 0 * 0} & e^{-ifrac{2 pi}{4}* 0 * 1} & e^{-ifrac{2 pi}{4}* 0 * 2} & e^{-ifrac{2 pi}{4}* 0 * 3}
e^{-ifrac{2 pi}{4}* 1 * 0} & e^{-ifrac{2 pi}{4}* 1 * 1} & e^{-ifrac{2 pi}{4}* 1 * 2} & e^{-ifrac{2 pi}{4}* 1 * 3}
e^{-ifrac{2 pi}{4}* 2 * 0} & e^{-ifrac{2 pi}{4}* 2 * 1} & e^{-ifrac{2 pi}{4}* 2 * 2} & e^{-ifrac{2 pi}{4}* 2 * 3}
e^{-ifrac{2 pi}{4}* 3 * 0} & e^{-ifrac{2 pi}{4}* 3 * 1} & e^{-ifrac{2 pi}{4}* 3 * 2} & e^{-ifrac{2 pi}{4}* 3 * 3}
end{bmatrix}
] {#eq-dft-3}

Or, evaluating the expressions:

[
mathbf{W}_4
=
begin{bmatrix}
1 & 1 & 1 & 1
1 & -i & -1 & i
1 & -1 & 1 & -1
1 & i & -1 & -i
end{bmatrix}
]

With that modification, the code appears to be like much more elegant:

dft_vec <- operate(x) {
  n_samples <- size(x)

  n <- torch_arange(0, n_samples - 1)$unsqueeze(1)
  ok <- torch_arange(0, n_samples - 1)$unsqueeze(2)

  mat_k_m <- torch_exp(-1i * 2 * pi / n_samples * ok * n)

  torch_matmul(mat_k_m, x$to(dtype = torch_cfloat()))
}

As you possibly can simply confirm, the end result is identical.

Thanks for studying!

Photograph by Trac Vu on Unsplash