mirror of https://code.videolan.org/videolan/vlc
1179 lines
40 KiB
Python
1179 lines
40 KiB
Python
# Lossy compression algorithms very often make use of DCT or DFT calculations,
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# or variations of these calculations. This file is intended to be a short
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# reference about classical DCT and DFT algorithms.
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import math
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import cmath
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pi = math.pi
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sin = math.sin
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cos = math.cos
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sqrt = math.sqrt
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def exp_j (alpha):
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return cmath.exp (alpha * 1j)
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def conjugate (c):
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c = c + 0j
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return c.real - 1j * c.imag
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def vector (N):
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return [0j] * N
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# Let us start withthe canonical definition of the unscaled DFT algorithm :
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# (I can not draw sigmas in a text file so I'll use python code instead) :)
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def W (k, N):
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return exp_j ((-2*pi*k)/N)
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def unscaled_DFT (N, input, output):
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for o in range(N): # o is output index
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output[o] = 0
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for i in range(N):
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output[o] = output[o] + input[i] * W (i*o, N)
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# This algorithm takes complex input and output. There are N*N complex
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# multiplications and N*(N-1) complex additions.
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# Of course this algorithm is an extremely naive implementation and there are
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# some ways to use the trigonometric properties of the coefficients to find
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# some decompositions that can accelerate the calculation by several orders
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# of magnitude... This is a well known and studied problem. One of the
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# available explanations of this process is at this url :
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# www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fft.html
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# Let's start with the radix-2 decimation-in-time algorithm :
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def unscaled_DFT_radix2_time (N, input, output):
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even_input = vector(N/2)
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odd_input = vector(N/2)
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even_output = vector(N/2)
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odd_output = vector(N/2)
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for i in range(N/2):
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even_input[i] = input[2*i]
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odd_input[i] = input[2*i+1]
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unscaled_DFT (N/2, even_input, even_output)
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unscaled_DFT (N/2, odd_input, odd_output)
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for i in range(N/2):
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odd_output[i] = odd_output[i] * W (i, N)
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for i in range(N/2):
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output[i] = even_output[i] + odd_output[i]
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output[i+N/2] = even_output[i] - odd_output[i]
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# This algorithm takes complex input and output.
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# We divide the DFT calculation into 2 DFT calculations of size N/2
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# We then do N/2 complex multiplies followed by N complex additions.
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# (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex
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# multiplies... we will skip 1 for i=0 and 1 for i=N/4. Also for i=N/8 and for
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# i=3*N/8 the W(i,N) values can be special-cased to implement the complex
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# multiplication using only 2 real additions and 2 real multiplies)
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# Also note that all the basic stages of this DFT algorithm are easily
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# reversible, so we can calculate the IDFT with the same complexity.
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# A varient of this is the radix-2 decimation-in-frequency algorithm :
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def unscaled_DFT_radix2_freq (N, input, output):
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even_input = vector(N/2)
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odd_input = vector(N/2)
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even_output = vector(N/2)
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odd_output = vector(N/2)
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for i in range(N/2):
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even_input[i] = input[i] + input[i+N/2]
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odd_input[i] = input[i] - input[i+N/2]
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for i in range(N/2):
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odd_input[i] = odd_input[i] * W (i, N)
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unscaled_DFT (N/2, even_input, even_output)
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unscaled_DFT (N/2, odd_input, odd_output)
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for i in range(N/2):
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output[2*i] = even_output[i]
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output[2*i+1] = odd_output[i]
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# Note that the decimation-in-time and the decimation-in-frequency varients
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# have exactly the same complexity, they only do the operations in a different
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# order.
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# Actually, if you look at the decimation-in-time varient of the DFT, and
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# reverse it to calculate an IDFT, you get something that is extremely close
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# to the decimation-in-frequency DFT algorithm...
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# The radix-4 algorithms are slightly more efficient : they take into account
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# the fact that with complex numbers, multiplications by j and -j are also
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# "free"... i.e. when you code them using real numbers, they translate into
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# a few data moves but no real operation.
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# Let's start with the radix-4 decimation-in-time algorithm :
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def unscaled_DFT_radix4_time (N, input, output):
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input_0 = vector(N/4)
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input_1 = vector(N/4)
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input_2 = vector(N/4)
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input_3 = vector(N/4)
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output_0 = vector(N/4)
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output_1 = vector(N/4)
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output_2 = vector(N/4)
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output_3 = vector(N/4)
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tmp_0 = vector(N/4)
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tmp_1 = vector(N/4)
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tmp_2 = vector(N/4)
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tmp_3 = vector(N/4)
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for i in range(N/4):
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input_0[i] = input[4*i]
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input_1[i] = input[4*i+1]
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input_2[i] = input[4*i+2]
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input_3[i] = input[4*i+3]
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unscaled_DFT (N/4, input_0, output_0)
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unscaled_DFT (N/4, input_1, output_1)
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unscaled_DFT (N/4, input_2, output_2)
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unscaled_DFT (N/4, input_3, output_3)
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for i in range(N/4):
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output_1[i] = output_1[i] * W (i, N)
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output_2[i] = output_2[i] * W (2*i, N)
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output_3[i] = output_3[i] * W (3*i, N)
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for i in range(N/4):
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tmp_0[i] = output_0[i] + output_2[i]
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tmp_1[i] = output_0[i] - output_2[i]
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tmp_2[i] = output_1[i] + output_3[i]
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tmp_3[i] = output_1[i] - output_3[i]
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for i in range(N/4):
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output[i] = tmp_0[i] + tmp_2[i]
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output[i+N/4] = tmp_1[i] - 1j * tmp_3[i]
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output[i+N/2] = tmp_0[i] - tmp_2[i]
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output[i+3*N/4] = tmp_1[i] + 1j * tmp_3[i]
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# This algorithm takes complex input and output.
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# We divide the DFT calculation into 4 DFT calculations of size N/4
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# We then do 3*N/4 complex multiplies followed by 2*N complex additions.
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# (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex
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# multiplies... we will skip 3 for i=0 and 1 for i=N/8. Also for i=N/8
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# the remaining W(i,N) and W(3*i,N) multiplies can be implemented using only
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# 2 real additions and 2 real multiplies. For i=N/16 and i=3*N/16 we can also
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# optimise the W(2*i/N) multiply this way.
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# If we wanted to do the same decomposition with one radix-2 decomposition
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# of size N and 2 radix-2 decompositions of size N/2, the total cost would be
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# N complex multiplies and 2*N complex additions. Thus we see that the
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# decomposition of one DFT calculation of size N into 4 calculations of size
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# N/4 using the radix-4 algorithm instead of the radix-2 algorithm saved N/4
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# complex multiplies...
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# The radix-4 decimation-in-frequency algorithm is similar :
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def unscaled_DFT_radix4_freq (N, input, output):
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input_0 = vector(N/4)
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input_1 = vector(N/4)
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input_2 = vector(N/4)
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input_3 = vector(N/4)
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output_0 = vector(N/4)
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output_1 = vector(N/4)
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output_2 = vector(N/4)
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output_3 = vector(N/4)
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tmp_0 = vector(N/4)
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tmp_1 = vector(N/4)
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tmp_2 = vector(N/4)
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tmp_3 = vector(N/4)
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for i in range(N/4):
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tmp_0[i] = input[i] + input[i+N/2]
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tmp_1[i] = input[i+N/4] + input[i+3*N/4]
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tmp_2[i] = input[i] - input[i+N/2]
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tmp_3[i] = input[i+N/4] - input[i+3*N/4]
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for i in range(N/4):
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input_0[i] = tmp_0[i] + tmp_1[i]
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input_1[i] = tmp_2[i] - 1j * tmp_3[i]
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input_2[i] = tmp_0[i] - tmp_1[i]
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input_3[i] = tmp_2[i] + 1j * tmp_3[i]
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for i in range(N/4):
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input_1[i] = input_1[i] * W (i, N)
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input_2[i] = input_2[i] * W (2*i, N)
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input_3[i] = input_3[i] * W (3*i, N)
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unscaled_DFT (N/4, input_0, output_0)
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unscaled_DFT (N/4, input_1, output_1)
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unscaled_DFT (N/4, input_2, output_2)
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unscaled_DFT (N/4, input_3, output_3)
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for i in range(N/4):
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output[4*i] = output_0[i]
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output[4*i+1] = output_1[i]
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output[4*i+2] = output_2[i]
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output[4*i+3] = output_3[i]
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# Once again the complexity is exactly the same as for the radix-4
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# decimation-in-time DFT algorithm, only the order of the operations is
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# different.
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# Now let us reorder the radix-4 algorithms in a different way :
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#def unscaled_DFT_radix4_time (N, input, output):
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# input_0 = vector(N/4)
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# input_1 = vector(N/4)
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# input_2 = vector(N/4)
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# input_3 = vector(N/4)
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# output_0 = vector(N/4)
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# output_1 = vector(N/4)
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# output_2 = vector(N/4)
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# output_3 = vector(N/4)
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# tmp_0 = vector(N/4)
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# tmp_1 = vector(N/4)
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# tmp_2 = vector(N/4)
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# tmp_3 = vector(N/4)
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#
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# for i in range(N/4):
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# input_0[i] = input[4*i]
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# input_2[i] = input[4*i+2]
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#
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# unscaled_DFT (N/4, input_0, output_0)
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# unscaled_DFT (N/4, input_2, output_2)
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#
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# for i in range(N/4):
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# output_2[i] = output_2[i] * W (2*i, N)
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#
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# for i in range(N/4):
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# tmp_0[i] = output_0[i] + output_2[i]
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# tmp_1[i] = output_0[i] - output_2[i]
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#
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# for i in range(N/4):
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# input_1[i] = input[4*i+1]
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# input_3[i] = input[4*i+3]
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#
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# unscaled_DFT (N/4, input_1, output_1)
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# unscaled_DFT (N/4, input_3, output_3)
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#
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# for i in range(N/4):
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# output_1[i] = output_1[i] * W (i, N)
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# output_3[i] = output_3[i] * W (3*i, N)
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#
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# for i in range(N/4):
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# tmp_2[i] = output_1[i] + output_3[i]
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# tmp_3[i] = output_1[i] - output_3[i]
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#
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# for i in range(N/4):
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# output[i] = tmp_0[i] + tmp_2[i]
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# output[i+N/4] = tmp_1[i] - 1j * tmp_3[i]
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# output[i+N/2] = tmp_0[i] - tmp_2[i]
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# output[i+3*N/4] = tmp_1[i] + 1j * tmp_3[i]
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# We didnt do anything here, only reorder the operations. But now, look at the
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# first part of this function, up to the calculations of tmp0 and tmp1 : this
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# is extremely similar to the radix-2 decimation-in-time algorithm ! or more
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# precisely, it IS the radix-2 decimation-in-time algorithm, with size N/2,
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# applied on a vector representing the even input coefficients, and giving
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# an output vector that is the concatenation of tmp0 and tmp1.
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# This is important to notice, because this means we can now choose to
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# calculate tmp0 and tmp1 using any DFT algorithm that we want, and we know
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# that some of them are more efficient than radix-2...
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# This leads us directly to the split-radix decimation-in-time algorithm :
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def unscaled_DFT_split_radix_time (N, input, output):
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even_input = vector(N/2)
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input_1 = vector(N/4)
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input_3 = vector(N/4)
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even_output = vector(N/2)
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output_1 = vector(N/4)
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output_3 = vector(N/4)
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tmp_0 = vector(N/4)
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tmp_1 = vector(N/4)
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for i in range(N/2):
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even_input[i] = input[2*i]
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for i in range(N/4):
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input_1[i] = input[4*i+1]
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input_3[i] = input[4*i+3]
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unscaled_DFT (N/2, even_input, even_output)
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unscaled_DFT (N/4, input_1, output_1)
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unscaled_DFT (N/4, input_3, output_3)
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for i in range(N/4):
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output_1[i] = output_1[i] * W (i, N)
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output_3[i] = output_3[i] * W (3*i, N)
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for i in range(N/4):
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tmp_0[i] = output_1[i] + output_3[i]
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tmp_1[i] = output_1[i] - output_3[i]
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for i in range(N/4):
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output[i] = even_output[i] + tmp_0[i]
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output[i+N/4] = even_output[i+N/4] - 1j * tmp_1[i]
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output[i+N/2] = even_output[i] - tmp_0[i]
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output[i+3*N/4] = even_output[i+N/4] + 1j * tmp_1[i]
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# This function performs one DFT of size N/2 and two of size N/4, followed by
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# N/2 complex multiplies and 3*N/2 complex additions.
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# (actually W(0,N) = 1 and W(N/4,N) = -j so we can skip a few of these complex
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# multiplies... we will skip 2 for i=0. Also for i=N/8 the W(i,N) and W(3*i,N)
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# multiplies can be implemented using only 2 real additions and 2 real
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# multiplies)
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# We can similarly define the split-radix decimation-in-frequency DFT :
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def unscaled_DFT_split_radix_freq (N, input, output):
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even_input = vector(N/2)
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input_1 = vector(N/4)
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input_3 = vector(N/4)
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even_output = vector(N/2)
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output_1 = vector(N/4)
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output_3 = vector(N/4)
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tmp_0 = vector(N/4)
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tmp_1 = vector(N/4)
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for i in range(N/2):
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even_input[i] = input[i] + input[i+N/2]
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for i in range(N/4):
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tmp_0[i] = input[i] - input[i+N/2]
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tmp_1[i] = input[i+N/4] - input[i+3*N/4]
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for i in range(N/4):
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input_1[i] = tmp_0[i] - 1j * tmp_1[i]
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input_3[i] = tmp_0[i] + 1j * tmp_1[i]
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for i in range(N/4):
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input_1[i] = input_1[i] * W (i, N)
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input_3[i] = input_3[i] * W (3*i, N)
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unscaled_DFT (N/2, even_input, even_output)
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unscaled_DFT (N/4, input_1, output_1)
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unscaled_DFT (N/4, input_3, output_3)
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for i in range(N/2):
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output[2*i] = even_output[i]
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for i in range(N/4):
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output[4*i+1] = output_1[i]
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output[4*i+3] = output_3[i]
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# The complexity is again the same as for the decimation-in-time varient.
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# Now let us now summarize our various algorithms for DFT decomposition :
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# radix-2 : DFT(N) -> 2*DFT(N/2) using N/2 multiplies and N additions
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# radix-4 : DFT(N) -> 4*DFT(N/2) using 3*N/4 multiplies and 2*N additions
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# split-radix : DFT(N) -> DFT(N/2) + 2*DFT(N/4) using N/2 muls and 3*N/2 adds
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# (we are always speaking of complex multiplies and complex additions... a
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# complex addition is implemented with 2 real additions, and a complex
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# multiply is implemented with either 2 adds and 4 muls or 3 adds and 3 muls,
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# so we will keep a separate count of these)
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# If we want to take into account the special values of W(i,N), we can remove
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# a few complex multiplies. Supposing N>=16 we can remove :
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# radix-2 : remove 2 complex multiplies, simplify 2 others
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# radix-4 : remove 4 complex multiplies, simplify 4 others
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# split-radix : remove 2 complex multiplies, simplify 2 others
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# This gives the following table for the complexity of a complex DFT :
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# N real additions real multiplies complex multiplies
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# 1 0 0 0
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# 2 4 0 0
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# 4 16 0 0
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# 8 52 4 0
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# 16 136 8 4
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# 32 340 20 16
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# 64 808 40 52
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# 128 1876 84 144
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# 256 4264 168 372
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# 512 9556 340 912
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# 1024 21160 680 2164
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# 2048 46420 1364 5008
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# 4096 101032 2728 11380
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# 8192 218452 5460 25488
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# 16384 469672 10920 56436
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# 32768 1004884 21844 123792
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# 65536 2140840 43688 269428
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# If we chose to implement complex multiplies with 3 real muls + 3 real adds,
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# then these results are consistent with the table at the end of the
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# www.cmlab.csie.ntu.edu.tw DFT tutorial that I mentionned earlier.
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# Now another important case for the DFT is the one where the inputs are
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# real numbers instead of complex ones. We have to find ways to optimize for
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# this important case.
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# If the DFT inputs are real-valued, then the DFT outputs have nice properties
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# too : output[0] and output[N/2] will be real numbers, and output[N-i] will
|
|
# be the conjugate of output[i] for i in 0...N/2-1
|
|
|
|
# Likewise, if the DFT inputs are purely imaginary numbers, then the DFT
|
|
# outputs will have special properties too : output[0] and output[N/2] will be
|
|
# purely imaginary, and output[N-i] will be the opposite of the conjugate of
|
|
# output[i] for i in 0...N/2-1
|
|
|
|
# We can use these properties to calculate two real-valued DFT at once :
|
|
|
|
def two_real_unscaled_DFT (N, input1, input2, output1, output2):
|
|
input = vector(N)
|
|
output = vector(N)
|
|
|
|
for i in range(N):
|
|
input[i] = input1[i] + 1j * input2[i]
|
|
|
|
unscaled_DFT (N, input, output)
|
|
|
|
output1[0] = output[0].real + 0j
|
|
output2[0] = output[0].imag + 0j
|
|
|
|
for i in range(N/2)[1:]:
|
|
output1[i] = 0.5 * (output[i] + conjugate(output[N-i]))
|
|
output2[i] = -0.5j * (output[i] - conjugate(output[N-i]))
|
|
|
|
output1[N-i] = conjugate(output1[i])
|
|
output2[N-i] = conjugate(output2[i])
|
|
|
|
output1[N/2] = output[N/2].real + 0j
|
|
output2[N/2] = output[N/2].imag + 0j
|
|
|
|
# This routine does a total of N-2 complex additions and N-2 complex
|
|
# multiplies by 0.5
|
|
|
|
# This routine can also be inverted to calculate the IDFT of two vectors at
|
|
# once if we know that the outputs will be real-valued.
|
|
|
|
|
|
# If we have only one real-valued DFT calculation to do, we can still cut this
|
|
# calculation in several parts using one of the decimate-in-time methods
|
|
# (so that the different parts are still real-valued)
|
|
|
|
# As with complex DFT calculations, the best method is to use a split radix.
|
|
# There are a lot of symetries in the DFT outputs that we can exploit to
|
|
# reduce the number of operations...
|
|
|
|
def real_unscaled_DFT_split_radix_time_1 (N, input, output):
|
|
even_input = vector(N/2)
|
|
even_output = vector(N/2)
|
|
input_1 = vector(N/4)
|
|
output_1 = vector(N/4)
|
|
input_3 = vector(N/4)
|
|
output_3 = vector(N/4)
|
|
tmp_0 = vector(N/4)
|
|
tmp_1 = vector(N/4)
|
|
|
|
for i in range(N/2):
|
|
even_input[i] = input[2*i]
|
|
|
|
for i in range(N/4):
|
|
input_1[i] = input[4*i+1]
|
|
input_3[i] = input[4*i+3]
|
|
|
|
unscaled_DFT (N/2, even_input, even_output)
|
|
# this is again a real DFT !
|
|
# we will only use even_output[i] for i in 0 ... N/4 included. we know that
|
|
# even_output[N/2-i] is the conjugate of even_output[i]... also we know
|
|
# that even_output[0] and even_output[N/4] are purely real.
|
|
|
|
unscaled_DFT (N/4, input_1, output_1)
|
|
unscaled_DFT (N/4, input_3, output_3)
|
|
# these are real DFTs too... with symetries in the outputs... once again
|
|
|
|
tmp_0[0] = output_1[0] + output_3[0] # real numbers
|
|
tmp_1[0] = output_1[0] - output_3[0] # real numbers
|
|
|
|
tmp__0 = (output_1[N/8] + output_3[N/8]) * sqrt(0.5) # real numbers
|
|
tmp__1 = (output_1[N/8] - output_3[N/8]) * sqrt(0.5) # real numbers
|
|
tmp_0[N/8] = tmp__1 - 1j * tmp__0 # real + 1j * real
|
|
tmp_1[N/8] = tmp__0 - 1j * tmp__1 # real + 1j * real
|
|
|
|
for i in range(N/8)[1:]:
|
|
output_1[i] = output_1[i] * W (i, N)
|
|
output_3[i] = output_3[i] * W (3*i, N)
|
|
|
|
tmp_0[i] = output_1[i] + output_3[i]
|
|
tmp_1[i] = output_1[i] - output_3[i]
|
|
|
|
tmp_0[N/4-i] = -1j * conjugate(tmp_1[i])
|
|
tmp_1[N/4-i] = -1j * conjugate(tmp_0[i])
|
|
|
|
output[0] = even_output[0] + tmp_0[0] # real numbers
|
|
output[N/4] = even_output[N/4] - 1j * tmp_1[0] # real + 1j * real
|
|
output[N/2] = even_output[0] - tmp_0[0] # real numbers
|
|
output[3*N/4] = even_output[N/4] + 1j * tmp_1[0] # real + 1j * real
|
|
|
|
for i in range(N/4)[1:]:
|
|
output[i] = even_output[i] + tmp_0[i]
|
|
output[i+N/4] = conjugate(even_output[N/4-i]) - 1j * tmp_1[i]
|
|
|
|
output[N-i] = conjugate(output[i])
|
|
output[3*N/4-i] = conjugate(output[i+N/4])
|
|
|
|
# This function performs one real DFT of size N/2 and two real DFT of size
|
|
# N/4, followed by 6 real additions, 2 real multiplies, 3*N/4-4 complex
|
|
# additions and N/4-2 complex multiplies.
|
|
|
|
|
|
# We can also try to combine the two real DFT of size N/4 into a single complex
|
|
# DFT :
|
|
|
|
def real_unscaled_DFT_split_radix_time_2 (N, input, output):
|
|
even_input = vector(N/2)
|
|
even_output = vector(N/2)
|
|
odd_input = vector(N/4)
|
|
odd_output = vector(N/4)
|
|
tmp_0 = vector(N/4)
|
|
tmp_1 = vector(N/4)
|
|
|
|
for i in range(N/2):
|
|
even_input[i] = input[2*i]
|
|
|
|
for i in range(N/4):
|
|
odd_input[i] = input[4*i+1] + 1j * input[4*i+3]
|
|
|
|
unscaled_DFT (N/2, even_input, even_output)
|
|
# this is again a real DFT !
|
|
# we will only use even_output[i] for i in 0 ... N/4 included. we know that
|
|
# even_output[N/2-i] is the conjugate of even_output[i]... also we know
|
|
# that even_output[0] and even_output[N/4] are purely real.
|
|
|
|
unscaled_DFT (N/4, odd_input, odd_output)
|
|
# but this one is a complex DFT so no special properties here
|
|
|
|
output_1 = odd_output[0].real
|
|
output_3 = odd_output[0].imag
|
|
tmp_0[0] = output_1 + output_3 # real numbers
|
|
tmp_1[0] = output_1 - output_3 # real numbers
|
|
|
|
output_1 = odd_output[N/8].real
|
|
output_3 = odd_output[N/8].imag
|
|
tmp__0 = (output_1 + output_3) * sqrt(0.5) # real numbers
|
|
tmp__1 = (output_1 - output_3) * sqrt(0.5) # real numbers
|
|
tmp_0[N/8] = tmp__1 - 1j * tmp__0 # real + 1j * real
|
|
tmp_1[N/8] = tmp__0 - 1j * tmp__1 # real + 1j * real
|
|
|
|
for i in range(N/8)[1:]:
|
|
output_1 = odd_output[i] + conjugate(odd_output[N/4-i])
|
|
output_3 = odd_output[i] - conjugate(odd_output[N/4-i])
|
|
|
|
output_1 = output_1 * 0.5 * W (i, N)
|
|
output_3 = output_3 * -0.5j * W (3*i, N)
|
|
|
|
tmp_0[i] = output_1 + output_3
|
|
tmp_1[i] = output_1 - output_3
|
|
|
|
tmp_0[N/4-i] = -1j * conjugate(tmp_1[i])
|
|
tmp_1[N/4-i] = -1j * conjugate(tmp_0[i])
|
|
|
|
output[0] = even_output[0] + tmp_0[0] # real numbers
|
|
output[N/4] = even_output[N/4] - 1j * tmp_1[0] # real + 1j * real
|
|
output[N/2] = even_output[0] - tmp_0[0] # real numbers
|
|
output[3*N/4] = even_output[N/4] + 1j * tmp_1[0] # real + 1j * real
|
|
|
|
for i in range(N/4)[1:]:
|
|
output[i] = even_output[i] + tmp_0[i]
|
|
output[i+N/4] = conjugate(even_output[N/4-i]) - 1j * tmp_1[i]
|
|
|
|
output[N-i] = conjugate(output[i])
|
|
output[3*N/4-i] = conjugate(output[i+N/4])
|
|
|
|
# This function performs one real DFT of size N/2 and one complex DFT of size
|
|
# N/4, followed by 6 real additions, 2 real multiplies, N-6 complex additions
|
|
# and N/4-2 complex multiplies.
|
|
|
|
# After comparing the performance, it turns out that for real-valued DFT, the
|
|
# version of the algorithm that subdivides the calculation into one real
|
|
# DFT of size N/2 and two real DFT of size N/4 is the most efficient one.
|
|
# The other version gives exactly the same number of multiplies and a few more
|
|
# real additions.
|
|
|
|
|
|
# Now we can also try the decimate-in-frequency method for a real-valued DFT.
|
|
# Using the split-radix algorithm, and by taking into account the symetries of
|
|
# the outputs :
|
|
|
|
def real_unscaled_DFT_split_radix_freq (N, input, output):
|
|
even_input = vector(N/2)
|
|
input_1 = vector(N/4)
|
|
even_output = vector(N/2)
|
|
output_1 = vector(N/4)
|
|
tmp_0 = vector(N/4)
|
|
tmp_1 = vector(N/4)
|
|
|
|
for i in range(N/2):
|
|
even_input[i] = input[i] + input[i+N/2]
|
|
|
|
for i in range(N/4):
|
|
tmp_0[i] = input[i] - input[i+N/2]
|
|
tmp_1[i] = input[i+N/4] - input[i+3*N/4]
|
|
|
|
for i in range(N/4):
|
|
input_1[i] = tmp_0[i] - 1j * tmp_1[i]
|
|
|
|
for i in range(N/4):
|
|
input_1[i] = input_1[i] * W (i, N)
|
|
|
|
unscaled_DFT (N/2, even_input, even_output)
|
|
# This is still a real-valued DFT
|
|
|
|
unscaled_DFT (N/4, input_1, output_1)
|
|
# But that one is a complex-valued DFT
|
|
|
|
for i in range(N/2):
|
|
output[2*i] = even_output[i]
|
|
|
|
for i in range(N/4):
|
|
output[4*i+1] = output_1[i]
|
|
output[N-1-4*i] = conjugate(output_1[i])
|
|
|
|
# I think this implementation is much more elegant than the decimate-in-time
|
|
# version ! It looks very much like the complex-valued version, all we had to
|
|
# do was remove one of the complex-valued internal DFT calls because we could
|
|
# deduce the outputs by using the symetries of the problem.
|
|
|
|
# As for performance, we did N real additions, N/4 complex multiplies (a bit
|
|
# less actually, because W(0,N) = 1 and W(N/8,N) is a "simple" multiply), then
|
|
# one real DFT of size N/2 and one complex DFT of size N/4.
|
|
|
|
# It turns out that even if the methods are so different, the number of
|
|
# operations is exactly the same as for the best of the two decimation-in-time
|
|
# methods that we tried.
|
|
|
|
|
|
# This gives us the following performance for real-valued DFT :
|
|
# N real additions real multiplies complex multiplies
|
|
# 2 2 0 0
|
|
# 4 6 0 0
|
|
# 8 20 2 0
|
|
# 16 54 4 2
|
|
# 32 140 10 8
|
|
# 64 342 20 26
|
|
# 128 812 42 72
|
|
# 256 1878 84 186
|
|
# 512 4268 170 456
|
|
# 1024 9558 340 1082
|
|
# 2048 21164 682 2504
|
|
# 4096 46422 1364 5690
|
|
# 8192 101036 2730 12744
|
|
# 16384 218454 5460 28218
|
|
# 32768 469676 10922 61896
|
|
# 65536 1004886 21844 134714
|
|
|
|
|
|
# As an example, this is an implementation of the real-valued DFT8 :
|
|
|
|
def DFT8 (input, output):
|
|
even_0 = input[0] + input[4]
|
|
even_1 = input[1] + input[5]
|
|
even_2 = input[2] + input[6]
|
|
even_3 = input[3] + input[7]
|
|
|
|
tmp_0 = even_0 + even_2
|
|
tmp_1 = even_0 - even_2
|
|
tmp_2 = even_1 + even_3
|
|
tmp_3 = even_1 - even_3
|
|
|
|
output[0] = tmp_0 + tmp_2
|
|
output[2] = tmp_1 - 1j * tmp_3
|
|
output[4] = tmp_0 - tmp_2
|
|
|
|
odd_0_r = input[0] - input[4]
|
|
odd_0_i = input[2] - input[6]
|
|
|
|
tmp_0 = input[1] - input[5]
|
|
tmp_1 = input[3] - input[7]
|
|
odd_1_r = (tmp_0 - tmp_1) * sqrt(0.5)
|
|
odd_1_i = (tmp_0 + tmp_1) * sqrt(0.5)
|
|
|
|
output[1] = (odd_0_r + odd_1_r) - 1j * (odd_0_i + odd_1_i)
|
|
output[5] = (odd_0_r - odd_1_r) - 1j * (odd_0_i - odd_1_i)
|
|
|
|
output[3] = conjugate(output[5])
|
|
output[6] = conjugate(output[2])
|
|
output[7] = conjugate(output[1])
|
|
|
|
|
|
# Also a basic implementation of the real-valued DFT4 :
|
|
|
|
def DFT4 (input, output):
|
|
tmp_0 = input[0] + input[2]
|
|
tmp_1 = input[0] - input[2]
|
|
tmp_2 = input[1] + input[3]
|
|
tmp_3 = input[1] - input[3]
|
|
|
|
output[0] = tmp_0 + tmp_2
|
|
output[1] = tmp_1 - 1j * tmp_3
|
|
output[2] = tmp_0 - tmp_2
|
|
output[3] = tmp_1 + 1j * tmp_3
|
|
|
|
|
|
# A similar idea might be used to calculate only the real part of the output
|
|
# of a complex DFT : we take an DFT algorithm for real inputs and complex
|
|
# outputs and we simply reverse it. The resulting algorithm will only work
|
|
# with inputs that satisfy the conjugaison rule (input[i] is the conjugate of
|
|
# input[N-i]) so we can do a first pass to modify the input so that it follows
|
|
# this rule. An example implementation is as follows (adapted from the
|
|
# unscaled_DFT_split_radix_time algorithm) :
|
|
|
|
def complex2real_unscaled_DFT_split_radix_time (N, input, output):
|
|
even_input = vector(N/2)
|
|
input_1 = vector(N/4)
|
|
even_output = vector(N/2)
|
|
output_1 = vector(N/4)
|
|
|
|
for i in range(N/2):
|
|
even_input[i] = input[2*i]
|
|
|
|
for i in range(N/4):
|
|
input_1[i] = input[4*i+1] + conjugate(input[N-1-4*i])
|
|
|
|
unscaled_DFT (N/2, even_input, even_output)
|
|
unscaled_DFT (N/4, input_1, output_1)
|
|
|
|
for i in range(N/4):
|
|
output_1[i] = output_1[i] * W (i, N)
|
|
|
|
for i in range(N/4):
|
|
output[i] = even_output[i] + output_1[i].real
|
|
output[i+N/4] = even_output[i+N/4] + output_1[i].imag
|
|
output[i+N/2] = even_output[i] - output_1[i].real
|
|
output[i+3*N/4] = even_output[i+N/4] - output_1[i].imag
|
|
|
|
# This algorithm does N/4 complex additions, N/4-1 complex multiplies
|
|
# (including one "simple" multiply for i=N/8), N real additions, one
|
|
# "complex-to-real" DFT of size N/2, and one complex DFT of size N/4.
|
|
# Also, in the complex DFT of size N/4, we do not care about the imaginary
|
|
# part of output_1[0], which in practice allows us to save one real addition.
|
|
|
|
# This gives us the following performance for complex DFT with real outputs :
|
|
# N real additions real multiplies complex multiplies
|
|
# 1 0 0 0
|
|
# 2 2 0 0
|
|
# 4 8 0 0
|
|
# 8 25 2 0
|
|
# 16 66 4 2
|
|
# 32 167 10 8
|
|
# 64 400 20 26
|
|
# 128 933 42 72
|
|
# 256 2126 84 186
|
|
# 512 4771 170 456
|
|
# 1024 10572 340 1082
|
|
# 2048 23201 682 2504
|
|
# 4096 50506 1364 5690
|
|
# 8192 109215 2730 12744
|
|
# 16384 234824 5460 28218
|
|
# 32768 502429 10922 61896
|
|
# 65536 1070406 21844 134714
|
|
|
|
|
|
# Now let's talk about the DCT algorithm. The canonical definition for it is
|
|
# as follows :
|
|
|
|
def C (k, N):
|
|
return cos ((k*pi)/(2*N))
|
|
|
|
def unscaled_DCT (N, input, output):
|
|
for o in range(N): # o is output index
|
|
output[o] = 0
|
|
for i in range(N): # i is input index
|
|
output[o] = output[o] + input[i] * C ((2*i+1)*o, N)
|
|
|
|
# This trivial algorithm uses N*N multiplications and N*(N-1) additions.
|
|
|
|
|
|
# One possible decomposition on this calculus is to use the fact that C (i, N)
|
|
# and C (2*N-i, N) are opposed. This can lead to this decomposition :
|
|
|
|
#def unscaled_DCT (N, input, output):
|
|
# even_input = vector (N)
|
|
# odd_input = vector (N)
|
|
# even_output = vector (N)
|
|
# odd_output = vector (N)
|
|
#
|
|
# for i in range(N/2):
|
|
# even_input[i] = input[i] + input[N-1-i]
|
|
# odd_input[i] = input[i] - input[N-1-i]
|
|
#
|
|
# unscaled_DCT (N, even_input, even_output)
|
|
# unscaled_DCT (N, odd_input, odd_output)
|
|
#
|
|
# for i in range(N/2):
|
|
# output[2*i] = even_output[2*i]
|
|
# output[2*i+1] = odd_output[2*i+1]
|
|
|
|
# Now the even part can easily be calculated : by looking at the C(k,N)
|
|
# formula, we see that the even part is actually an unscaled DCT of size N/2.
|
|
# The odd part looks like a DCT of size N/2, but the coefficients are
|
|
# actually C ((2*i+1)*(2*o+1), 2*N) instead of C ((2*i+1)*o, N).
|
|
|
|
# We use a trigonometric relation here :
|
|
# 2 * C ((a+b)/2, N) * C ((a-b)/2, N) = C (a, N) + C (b, N)
|
|
# Thus with a = (2*i+1)*o and b = (2*i+1)*(o+1) :
|
|
# 2 * C((2*i+1)*(2*o+1),2N) * C(2*i+1,2N) = C((2*i+1)*o,N) + C((2*i+1)*(o+1),N)
|
|
|
|
# This leads us to the Lee DCT algorithm :
|
|
|
|
def unscaled_DCT_Lee (N, input, output):
|
|
even_input = vector(N/2)
|
|
odd_input = vector(N/2)
|
|
even_output = vector(N/2)
|
|
odd_output = vector(N/2)
|
|
|
|
for i in range(N/2):
|
|
even_input[i] = input[i] + input[N-1-i]
|
|
odd_input[i] = input[i] - input[N-1-i]
|
|
|
|
for i in range(N/2):
|
|
odd_input[i] = odd_input[i] * (0.5 / C (2*i+1, N))
|
|
|
|
unscaled_DCT (N/2, even_input, even_output)
|
|
unscaled_DCT (N/2, odd_input, odd_output)
|
|
|
|
for i in range(N/2-1):
|
|
odd_output[i] = odd_output[i] + odd_output[i+1]
|
|
|
|
for i in range(N/2):
|
|
output[2*i] = even_output[i]
|
|
output[2*i+1] = odd_output[i];
|
|
|
|
# Notes about this algorithm :
|
|
|
|
# The algorithm can be easily inverted to calculate the IDCT instead :
|
|
# each of the basic stages are separately inversible...
|
|
|
|
# This function does N adds, then N/2 muls, then 2 recursive calls with
|
|
# size N/2, then N/2-1 adds again. If we apply it recursively, the total
|
|
# number of operations will be N*log2(N)/2 multiplies and N*(3*log2(N)/2-1) + 1
|
|
# additions. So this is much faster than the canonical algorithm.
|
|
|
|
# Some of the multiplication coefficients 0.5/cos(...) can get quite large.
|
|
# This means that a small error in the input will give a large error on the
|
|
# output... For a DCT of size N the biggest coefficient will be at i=N/2-1
|
|
# and it will be slighly more than N/pi which can be large for large N's.
|
|
|
|
# In the IDCT however, the multiplication coefficients for the reverse
|
|
# transformation are of the form 2*cos(...) so they can not get big and there
|
|
# is no accuracy problem.
|
|
|
|
# You can find another description of this algorithm at
|
|
# http://www.intel.com/drg/mmx/appnotes/ap533.htm
|
|
|
|
|
|
|
|
# Another idea is to observe that the DCT calculation can be made to look like
|
|
# the DFT calculation : C (k, N) is the real part of W (k, 4*N) or W (-k, 4*N).
|
|
# We can use this idea translate the DCT algorithm into a call to the DFT
|
|
# algorithm :
|
|
|
|
def unscaled_DCT_DFT (N, input, output):
|
|
DFT_input = vector (4*N)
|
|
DFT_output = vector (4*N)
|
|
|
|
for i in range(N):
|
|
DFT_input[2*i+1] = input[i]
|
|
#DFT_input[4*N-2*i-1] = input[i] # We could use this instead
|
|
|
|
unscaled_DFT (4*N, DFT_input, DFT_output)
|
|
|
|
for i in range(N):
|
|
output[i] = DFT_output[i].real
|
|
|
|
|
|
# We can then use our knowledge of the DFT calculation to optimize for this
|
|
# particular case. For example using the radix-2 decimation-in-time method :
|
|
|
|
#def unscaled_DCT_DFT (N, input, output):
|
|
# DFT_input = vector (2*N)
|
|
# DFT_output = vector (2*N)
|
|
#
|
|
# for i in range(N):
|
|
# DFT_input[i] = input[i]
|
|
# #DFT_input[2*N-1-i] = input[i] # We could use this instead
|
|
#
|
|
# unscaled_DFT (2*N, DFT_input, DFT_output)
|
|
#
|
|
# for i in range(N):
|
|
# DFT_output[i] = DFT_output[i] * W (i, 4*N)
|
|
#
|
|
# for i in range(N):
|
|
# output[i] = DFT_output[i].real
|
|
|
|
# This leads us to the AAN implementation of the DCT algorithm : if we set
|
|
# both DFT_input[i] and DFT_input[2*N-1-i] to input[i], then the imaginary
|
|
# parts of W(2*i+1) and W(-2*i-1) will compensate, and output_DFT[i] will
|
|
# already be a real after the multiplication by W(i,4*N). Which means that
|
|
# before the multiplication, it is the product of a real number and W(-i,4*N).
|
|
# This leads to the following code, called the AAN algorithm :
|
|
|
|
def unscaled_DCT_AAN (N, input, output):
|
|
DFT_input = vector (2*N)
|
|
DFT_output = vector (2*N)
|
|
|
|
for i in range(N):
|
|
DFT_input[i] = input[i]
|
|
DFT_input[2*N-1-i] = input[i]
|
|
|
|
symetrical_unscaled_DFT (2*N, DFT_input, DFT_output)
|
|
|
|
for i in range(N):
|
|
output[i] = DFT_output[i].real * (0.5 / C (i, N))
|
|
|
|
# Notes about the AAN algorithm :
|
|
|
|
# The cost of this function is N real multiplies and a DFT of size 2*N. The
|
|
# DFT to calculate has special properties : the inputs are real and symmetric.
|
|
# Also, we only need to calculate the real parts of the N first DFT outputs.
|
|
# We can try to take advantage of all that.
|
|
|
|
# We can invert this algorithm to calculate the IDCT. The final multiply
|
|
# stage is trivially invertible. The DFT stage is invertible too, but we have
|
|
# to take into account the special properties of this particular DFT for that.
|
|
|
|
# Once again we have to take care of numerical precision for the DFT : the
|
|
# output coefficients can get large, so that a small error in the input will
|
|
# give a large error on the output... For a DCT of size N the biggest
|
|
# coefficient will be at i=N/2-1 and it will be slightly more than N/pi
|
|
|
|
# You can find another description of this algorithm at this url :
|
|
# www.cmlab.csie.ntu.edu.tw/cml/dsp/training/coding/transform/fastdct.html
|
|
# (It is the same server where we already found a description of the fast DFT)
|
|
|
|
|
|
# To optimize the DFT calculation, we can take a lot of specific things into
|
|
# account : the input is real and symetric, and we only care about the real
|
|
# part of the output. Also, we only care about the N first output coefficients,
|
|
# but that one does not save operations actually, because the other
|
|
# coefficients are the conjugates of the ones we look anyway.
|
|
|
|
# One useful way to use the symetry of the input is to use the radix-2
|
|
# decimation-in-frequency algorithm. We can write a version of
|
|
# unscaled_DFT_radix2_freq for the case where the input is symetrical :
|
|
# we have removed a few additions in the first stages because even_input
|
|
# is symetrical and odd_input is antisymetrical. Also, we have modified the
|
|
# odd_input vector so that the second half of it is set to zero and the real
|
|
# part of the DFT output is not modified. After that modification, the second
|
|
# part of the odd_input was null so we used the radix-2 decimation-in-frequency
|
|
# again on the odd DFT. Also odd_output is symetrical because input is real...
|
|
|
|
def symetrical_unscaled_DFT (N, input, output):
|
|
even_input = vector(N/2)
|
|
odd_tmp = vector(N/2)
|
|
odd_input = vector(N/2)
|
|
even_output = vector(N/2)
|
|
odd_output = vector(N/2)
|
|
|
|
for i in range(N/4):
|
|
even_input[N/2-i-1] = even_input[i] = input[i] + input[N/2-1-i]
|
|
|
|
for i in range(N/4):
|
|
odd_tmp[i] = input[i] - input[N/2-1-i]
|
|
|
|
odd_input[0] = odd_tmp[0]
|
|
for i in range(N/4)[1:]:
|
|
odd_input[i] = (odd_tmp[i] + odd_tmp[i-1]) * W (i, N)
|
|
|
|
unscaled_DFT (N/2, even_input, even_output)
|
|
# symetrical real inputs, real outputs
|
|
|
|
unscaled_DFT (N/4, odd_input, odd_output)
|
|
# complex inputs, real outputs
|
|
|
|
for i in range(N/2):
|
|
output[2*i] = even_output[i]
|
|
|
|
for i in range(N/4):
|
|
output[N-1-4*i] = output[4*i+1] = odd_output[i]
|
|
|
|
# This procedure takes 3*N/4-1 real additions and N/2-3 real multiplies,
|
|
# followed by another symetrical real DFT of size N/2 and a "complex to real"
|
|
# DFT of size N/4.
|
|
|
|
# We thus get the following performance results :
|
|
# N real additions real multiplies complex multiplies
|
|
# 1 0 0 0
|
|
# 2 0 0 0
|
|
# 4 2 0 0
|
|
# 8 9 1 0
|
|
# 16 28 6 0
|
|
# 32 76 21 0
|
|
# 64 189 54 2
|
|
# 128 451 125 10
|
|
# 256 1042 270 36
|
|
# 512 2358 565 108
|
|
# 1024 5251 1158 294
|
|
# 2048 11557 2349 750
|
|
# 4096 25200 4734 1832
|
|
# 8192 54544 9509 4336
|
|
# 16384 117337 19062 10026
|
|
# 32768 251127 38173 22770
|
|
# 65536 535102 76398 50988
|
|
|
|
|
|
# We thus get a better performance with the AAN DCT algorithm than with the
|
|
# Lee DCT algorithm : we can do a DCT of size 32 with 189 additions, 54+32 real
|
|
# multiplies, and 2 complex multiplies. The Lee algorithm would have used 209
|
|
# additions and 80 multiplies. With the AAN algorithm, we also have the
|
|
# advantage that a big number of the multiplies are actually grouped at the
|
|
# output stage of the algorithm, so if we want to do a DCT followed by a
|
|
# quantization stage, we will be able to group the multiply of the output with
|
|
# the multiply of the quantization stage, thus saving 32 more operations. In
|
|
# the mpeg audio layer 1 or 2 processing, we can also group the multiply of the
|
|
# output with the multiply of the convolution stage...
|
|
|
|
# Another source code for the AAN algorithm (implemented on 8 points, and
|
|
# without all of the explanations) can be found at this URL :
|
|
# http://developer.intel.com/drg/pentiumII/appnotes/aan_org.c . This
|
|
# implementation uses 28 adds and 6+8 muls instead of 29 adds and 5+8 muls -
|
|
# the difference is that in the symetrical_unscaled_DFT procedure, they noticed
|
|
# how odd_input[i] and odd_input[N/4-i] will be combined at the start of the
|
|
# complex-to-real DFT and they took advantage of this to convert 2 real adds
|
|
# and 4 real muls into one complex multiply.
|
|
|
|
|
|
# TODO : write about multi-dimentional DCT
|
|
|
|
|
|
# TEST CODE
|
|
|
|
def dump (vector):
|
|
str = ""
|
|
for i in range(len(vector)):
|
|
if i:
|
|
str = str + ", "
|
|
vector[i] = vector[i] + 0j
|
|
realstr = "%+.4f" % vector[i].real
|
|
imagstr = "%+.4fj" % vector[i].imag
|
|
if (realstr == "-0.0000"):
|
|
realstr = "+0.0000"
|
|
if (imagstr == "-0.0000j"):
|
|
imagstr = "+0.0000j"
|
|
str = str + realstr #+ imagstr
|
|
return "[%s]" % str
|
|
|
|
import whrandom
|
|
|
|
def test(N):
|
|
input = vector(N)
|
|
output = vector(N)
|
|
verify = vector(N)
|
|
|
|
for i in range(N):
|
|
input[i] = whrandom.random() + 1j * whrandom.random()
|
|
|
|
unscaled_DFT (N, input, output)
|
|
unscaled_DFT (N, input, verify)
|
|
|
|
if (dump(output) != dump(verify)):
|
|
print dump(output)
|
|
print dump(verify)
|
|
|
|
#test (64)
|
|
|
|
|
|
# PERFORMANCE ANALYSIS CODE
|
|
|
|
def display (table):
|
|
N = 1
|
|
print "#\tN\treal additions\treal multiplies\tcomplex multiplies"
|
|
while table.has_key(N):
|
|
print "#%8d%16d%16d%16d" % (N, table[N][0], table[N][1], table[N][2])
|
|
N = 2*N
|
|
print
|
|
|
|
best_complex_DFT = {}
|
|
|
|
def complex_DFT (max_N):
|
|
best_complex_DFT[1] = (0,0,0)
|
|
best_complex_DFT[2] = (4,0,0)
|
|
best_complex_DFT[4] = (16,0,0)
|
|
N = 8
|
|
while (N<=max_N):
|
|
# best method = split radix
|
|
best2 = best_complex_DFT[N/2]
|
|
best4 = best_complex_DFT[N/4]
|
|
best_complex_DFT[N] = (best2[0] + 2*best4[0] + 3*N + 4,
|
|
best2[1] + 2*best4[1] + 4,
|
|
best2[2] + 2*best4[2] + N/2 - 4)
|
|
N = 2*N
|
|
|
|
best_real_DFT = {}
|
|
|
|
def real_DFT (max_N):
|
|
best_real_DFT[1] = (0,0,0)
|
|
best_real_DFT[2] = (2,0,0)
|
|
best_real_DFT[4] = (6,0,0)
|
|
N = 8
|
|
while (N<=max_N):
|
|
# best method = split radix decimate-in-frequency
|
|
best2 = best_real_DFT[N/2]
|
|
best4 = best_complex_DFT[N/4]
|
|
best_real_DFT[N] = (best2[0] + best4[0] + N + 2,
|
|
best2[1] + best4[1] + 2,
|
|
best2[2] + best4[2] + N/4 - 2)
|
|
N = 2*N
|
|
|
|
best_complex2real_DFT = {}
|
|
|
|
def complex2real_DFT (max_N):
|
|
best_complex2real_DFT[1] = (0,0,0)
|
|
best_complex2real_DFT[2] = (2,0,0)
|
|
best_complex2real_DFT[4] = (8,0,0)
|
|
N = 8
|
|
while (N<=max_N):
|
|
best2 = best_complex2real_DFT[N/2]
|
|
best4 = best_complex_DFT[N/4]
|
|
best_complex2real_DFT[N] = (best2[0] + best4[0] + 3*N/2 + 1,
|
|
best2[1] + best4[1] + 2,
|
|
best2[2] + best4[2] + N/4 - 2)
|
|
N = 2*N
|
|
|
|
best_real_symetric_DFT = {}
|
|
|
|
def real_symetric_DFT (max_N):
|
|
best_real_symetric_DFT[1] = (0,0,0)
|
|
best_real_symetric_DFT[2] = (0,0,0)
|
|
best_real_symetric_DFT[4] = (2,0,0)
|
|
N = 8
|
|
while (N<=max_N):
|
|
best2 = best_real_symetric_DFT[N/2]
|
|
best4 = best_complex2real_DFT[N/4]
|
|
best_real_symetric_DFT[N] = (best2[0] + best4[0] + 3*N/4 - 1,
|
|
best2[1] + best4[1] + N/2 - 3,
|
|
best2[2] + best4[2])
|
|
N = 2*N
|
|
|
|
complex_DFT (65536)
|
|
real_DFT (65536)
|
|
complex2real_DFT (65536)
|
|
real_symetric_DFT (65536)
|
|
|
|
|
|
print "complex DFT"
|
|
display (best_complex_DFT)
|
|
|
|
print "real DFT"
|
|
display (best_real_DFT)
|
|
|
|
print "complex2real DFT"
|
|
display (best_complex2real_DFT)
|
|
|
|
print "real symetric DFT"
|
|
display (best_real_symetric_DFT)
|