Posit AI Weblog: Discrete Fourier Remodel

Word: 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 partly 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 manner as was potential to me, solid a light-weight on what’s behind the magic; it additionally exhibits how, surprisingly, you’ll be able to code the DFT in merely half a dozen traces. 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 much more materials than may sensibly match right into a weblog put up; subsequently, please think about what follows extra as a “teaser” than a totally fledged article.

Within the sciences, the Fourier Remodel is nearly in all places. Said very usually, it converts knowledge from one illustration to a different, with none lack of data (if finished appropriately, that’s.) When you use torch, it’s only a operate name away: torch_fft_fft() goes a method, torch_fft_ifft() the opposite. For the person, that’s handy – you “simply” must know easy methods to interpret the outcomes. Right here, I wish to assist with that. We begin with an instance operate name, taking part in 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 potential instance sign, a pure cosine that performs one revolution over the whole sampling interval.

Place to begin: A cosine of frequency 1

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

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

Right here (x) values progress over time (or area), and (okay) 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 (okay) 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 affirm this did what it was speculated to:

df <- knowledge.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 assorted frequencies current within the sign. The variety of frequencies thought-about will equal the variety of sampling factors: So (X) will probably 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 would name torch_fft_rfft() as a substitute, which yields “nicer” (within the sense of shorter) vectors to work with. Right here although, I wish 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 wanting 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, particularly, that at (okay = 1). This matches (and it higher needed to) the best way we constructed the sign: particularly, as engaging in a single revolution over the whole sampling interval.

Since, in concept, each coefficient may 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 replicate the respective actual components.

Lastly, there’s the section, indicating a potential shift of the sign (a pure cosine is unshifted). In torch, we’ve torch_angle() complementing torch_abs(), however we have to take into consideration roundoff error right here. We all know that in every however a single case, the true and imaginary components are each precisely zero; however attributable to 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 in all these “faux non-zeroes” and divide it by one other, as occurs within the angle calculation, huge values may end up. To forestall this from occurring, our customized implementation rounds each inputs earlier than triggering the division.

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

as.numeric(section(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 isn’t a section shift within the sign.

Let’s visualize what we discovered.

create_plot <- operate(x, y, amount) {
  df <- knowledge.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,
  section(Ft),
  "section"
)

p_real + p_imag + p_magnitude + p_phase

It’s honest to say that we’ve no cause 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, after 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 possibilities to satisfy your expectations, pricey reader, should be very near zero. Nonetheless, I wish to take the danger. When you’re an skilled on these items, you’ll anyway be simply scanning the textual content, looking for items of torch code. When you’re reasonably accustomed to the DFT, you should still like being reminded of its interior workings. And – most significantly – should you’re relatively new, and even utterly new, to this subject, you’ll hopefully take away (at the very least) one factor: that what looks like 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 features – 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 (okay) operating by means of the idea vectors, they are often written:

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

Like (okay), (n) runs from (0) to (N-1). To know what these foundation vectors are doing, it’s useful to briefly swap 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 will characterize these 4 foundation vectors when it comes to their “velocity”: how briskly they transfer across the unit circle. To do that, we merely have 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 cut-off dates. Which means a single “replace of place”, we will see how briskly the vector is transferring in a single time step.

Trying 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 one more step, and it will 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 manner, it finally ends up finishing two revolutions total. 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 best way we wrote the instance sign? Right here it’s once more:

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

If we handle to signify this operate when it comes to the idea vectors (mathbf{w}^{kn}_N = e^{ifrac{2 pi}{N}okay n}), the interior product between the operate and every foundation vector will probably be both zero (the “default”) or a a number of of 1 (in case the operate has a element matching the idea 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 method, 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 (okay)th Fourier coefficient is obtained by projecting the sign onto foundation vector (okay).

Because of the orthogonality of the idea vectors, solely two coefficients won’t 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 idea 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 closing 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, wanting again at what torch_fft_fft() gave us, we see we have been in a position to arrive on the similar outcome. And we’ve realized one thing alongside the best way.

So long as we stick with alerts composed of a number of foundation vectors, we will compute the DFT on this manner. On the finish of the chapter, we’ll develop code that can 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 wish to discover:

• What would occur if frequencies modified – say, a melody have been sung at the next pitch?

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

• What about section – 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 outcome ourselves.

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

Various frequency

Assume we quadrupled the frequency, giving us a sign that appeared 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 probably 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 outcome.

Now, let’s be certain that our evaluation is right. The next code snippet accommodates 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)  p_phase)


plot_ft(x)

This does certainly affirm our calculations.

A particular case arises when sign frequency rises to the very 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 seem like 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 here 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 after we differ amplitude. For instance, say the sign will get twice as loud. Now, there will probably 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)

Up to now, we’ve not as soon as seen a coefficient with non-zero imaginary half. To vary this, we add in section.

Including section

Altering the section 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 – start line of the sign.)

Now assume we shift the sign ahead by (frac{pi}{2}). Then the height we have 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 (okay=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 wish 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 section shift.

Now, we’re going to calculate the DFT for a shifted model of our instance sign. However should 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 traditional, 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 section 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 have a look at a wave that has greater than a single sinusoidal element.

Superposition of sinusoids

The sign we assemble should still be expressed when it comes to the idea 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 parts, and assemble the outcomes. With none calculation, nonetheless, there’s fairly a number of issues we will say:

• For the reason that sign consists of two pure cosines and one pure sine, there will probably be 4 coefficients with non-zero actual components, and two with non-zero imaginary components. The latter will probably be complicated conjugates of one another.
• From the best way 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 outcome 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 can we calculate the DFT for much less handy alerts?

Coding the DFT

Fortuitously, we already know what needs to be finished. We wish to venture the sign onto every of the idea vectors. In different phrases, we’ll be computing a bunch of interior merchandise. Logic-wise, nothing modifications: The one distinction is that normally, it won’t be potential to signify the sign when it comes to just some foundation vectors, like we did earlier than. Thus, all projections will truly should 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 concept is: To acquire a coefficient, venture the sign onto the corresponding foundation vector.

To implement that concept, we have to create the idea 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 purpose:

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 (okay in 0:(n_samples - 1)) {
    w_k <- torch_exp(-1i * 2 * pi / n_samples * okay * 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 examine 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 – should you look again – the outcomes are the identical.

Above, did I say “little code”? In reality, a loop just isn’t even wanted. As a substitute of working with the idea 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 probably be (N) of them. The columns correspond to positions (0) to (N-1); there will probably 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)$unsqueeze(1)
  okay <- torch_arange(0, n_samples - 1)$unsqueeze(2)

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

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

As you’ll be able to simply confirm, the outcome is identical.

Thanks for studying!

Picture by Trac Vu on Unsplash