RStudio AI Weblog: Discrete Fourier Rework

[ad_1]

Word: This publish 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 Rework (DFT), and is positioned partly three. Half three is devoted to scientific computation past deep studying.
There are two chapters on the Fourier Rework. The primary strives to, in as “verbal” and lucid a approach as was doable to me, forged a lightweight 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 Rework, 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 publish; due to this fact, please take into account what follows extra as a “teaser” than a totally fledged article.

Within the sciences, the Fourier Rework is nearly all over the place. Acknowledged very typically, it converts information from one illustration to a different, with none lack of data (if completed accurately, that’s.) In the event you use torch, it’s only a perform name away: torch_fft_fft() goes a method, torch_fft_ifft() the opposite. For the consumer, that’s handy – you “simply” have to know interpret the outcomes. Right here, I wish to assist with that. We begin with an instance perform 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 doable instance sign, a pure cosine that performs one revolution over the whole sampling interval.

Start line: A cosine of frequency 1

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

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

Right here (x) values progress over time (or house), 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 rapidly verify this did what it was alleged 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()
Pure cosine that accomplishes one revolution over the complete sample period (64 samples).

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 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 would name torch_fft_rfft() as an alternative, 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 completed 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 will 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, particularly, that at (okay = 1). This matches (and it higher needed to) the best way we constructed the sign: particularly, as conducting a single revolution over the whole sampling interval.

Since, in principle, each coefficient may have non-zero actual and imaginary components, usually 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 doable shift of the sign (a pure cosine is unshifted). In torch, we’ve torch_angle() complementing torch_abs(), however we have to keep 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 result of finite precision in how numbers are offered in a pc, the precise values will usually not be zero. As an alternative, they’ll be very small. If we take considered one of these “pretend non-zeroes” and divide it by one other, as occurs within the angle calculation, large values may result. To stop this from taking place, our customized implementation rounds each inputs earlier than triggering the division.

section <- perform(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 <- perform(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,
  section(Ft),
  "section"
)

p_real + p_imag + p_magnitude + p_phase
Real parts, imaginary parts, magnitudes and phases of the Fourier coefficients, obtained on a pure cosine that performs a single revolution over the sampling period. Imaginary parts as well as phases are all zero.

It’s honest to say that we’ve no cause to doubt what torch_fft_fft() has completed. However with a pure sinusoid like this, we are able to 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 Rework, and an viewers who I think about to range broadly on a dimension of math and sciences schooling, my probabilities to satisfy your expectations, expensive reader, should be very near zero. Nonetheless, I wish to take the chance. In the event you’re an skilled on these items, you’ll anyway be simply scanning the textual content, searching for items of torch code. In the event you’re reasonably aware of the DFT, you should still like being reminded of its interior workings. And – most significantly – in the event you’re quite new, and even utterly new, to this subject, you’ll hopefully take away (no less than) one factor: that what looks as if one of many best wonders of the universe (assuming there’s a actuality someway similar to what goes on in our minds) might be a surprise, however neither “magic” nor a factor reserved to the initiated.

In a nutshell, the Fourier Rework is a foundation transformation. Within the case of the DFT – the Discrete Fourier Rework, the place time and frequency representations each are finite vectors, not capabilities – the brand new foundation appears 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 way of the premise 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 change to a shorter sampling interval, (N = 4), say. If we accomplish that, 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 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 by way of their “velocity”: 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 totally different cut-off dates. Because of this a single “replace of place”, we are able to see how briskly the vector is shifting 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 could be again the place it began. That’s one revolution in 4 steps, or a step dimension of (frac{pi}{2}). Then (mathbf{w}^{2n}_N) goes at double that tempo, shifting 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 dimension 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 characterize this perform by way of the premise vectors (mathbf{w}^{kn}_N = e^{ifrac{2 pi}{N}okay n}), the interior product between the perform and every foundation vector will likely be both zero (the “default”) or a a number of of 1 (in case the perform has a element 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 system, 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 premise 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 perform 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 remaining 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 identical end result. And we’ve realized one thing alongside the best way.

So long as we stick with indicators composed of a number of foundation vectors, we are able to compute the DFT on this approach. On the finish of the chapter, we’ll develop code that may work for all indicators, however first, let’s see if we are able to 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 had been sung at the next pitch?

  • What about amplitude modifications – say, the music had 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 end result ourselves.

And at last, we’ll see how complicated sinusoids, made up of various parts, can nonetheless be analyzed on this approach, supplied they are often expressed by way of 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 are able to categorical 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 get hold of

[
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 identical end result.

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

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

plot_ft <- perform(x)  p_phase)


plot_ft(x)
A pure cosine that performs four revolutions over the sampling period, and its DFT. Imaginary parts and phases are still are zero.

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 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)
A pure cosine that performs thirty-two revolutions over the sampling period, and its DFT. This is the highest frequency where, given sixty-four sample points, no aliasing will occur. Imaginary parts and phases still zero.

Various amplitude

Now, let’s take into consideration what occurs once we range 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)
Pure cosine with four revolutions over the sampling period, and doubled amplitude. Imaginary parts and phases still zero.

Thus far, 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 perform 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 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 (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 get hold of 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 adverse section 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 like this:

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

First, we categorical it by way of 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 not equivalent. As an alternative, 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 verify our calculation utilizing torch_fft_fft().

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

plot_ft(x)
Delaying a pure cosine wave by pi/2 yields a pure sine wave. Now the real parts of all coefficients are zero; instead, non-zero imaginary values are appearing. The phase shift at those positions is pi/2.

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 take a look at a wave that has greater than a single sinusoidal element.

Superposition of sinusoids

The sign we assemble should still be expressed by way of the premise vectors, however it’s not a pure sinusoid. As an alternative, 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 totally 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 couple of issues we are able to 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 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 end result from multiplying with (frac{64}{2}) the scaling elements obtained for the person sinusoids.

Let’s verify:

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)
Superposition of pure sinusoids, and its DFT.

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

Coding the DFT

Luckily, we already know what needs to be completed. We wish to challenge 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 on the whole, it won’t be doable to characterize the sign by way of only a few foundation vectors, like we did earlier than. Thus, all projections will truly must 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 applied. 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, challenge 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 completed in a loop. Surprisingly little code is required to perform the objective:

dft <- perform(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 are able to 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 – in the event you look again – the outcomes are the identical.

Above, did I say “little code”? Actually, a loop isn’t even wanted. As an alternative of working with the premise vectors one-by-one, we are able to 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 much more elegant:

dft_vec <- perform(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 end result is identical.

Thanks for studying!

Photograph by Trac Vu on Unsplash

[ad_2]

Leave a Reply

Your email address will not be published. Required fields are marked *