The knowledge of the phase of STFT is a prerequisite for a successful signal reconstruction. However, the phase might be lost or no longer applicable depending on the kind of processing involved.
We propose a real-time spectrogram inversion algorithm based on the relationship of the gradients of the phase and the logarithm of the magnitude and on the gradient integration theorem. We present a detailed comparison with the state-of-the-art phase reconstruction algorithms.
The following achive ltfatnote043.zip contains scripts reproducing figures and tables from the paper.
Please note that LTFAT toolbox (version>=2.1.2, available here) and PHASERET (version>=0.1.0, available here) must be installed in order to run the scripts.
The following table shows error in dB for all 70 samples from the EBU SQAM database for the proposed algorithm RTPGHI and for the SPSI and RTISI-LA algorithms. The playback can be started by selecting one of the gray cells. All the files were compressed by oggenc from vorbis-tools 1.4.0 with the default settings. Your browser must support HTML5 audio player and it must be able to decode ogg files. Alternativelly, the file path is shown below the player and it can be downloaded by Save Link As ...
Table 3: Algorithm comparison a=256, extended Loaded file: None |
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Trunc. Gauss | Hann | Hamming | Blackman | |||||||||||||||||
SPSI | RTPGHI(0) | RTISI(0) | RTPGHI(1) | RTISI(1) | SPSI | RTPGHI(0) | RTISI(0) | RTPGHI(1) | RTISI(1) | SPSI | RTPGHI(0) | RTISI(0) | RTPGHI(1) | RTISI(1) | SPSI | RTPGHI(0) | RTISI(0) | RTPGHI(1) | RTISI(1) | |
01 Sine wave, 1 kHz | -38.31 | -49.48 | -36.66 | -49.94 | -35.91 | -35.73 | -40.98 | -36.86 | -41.48 | -31.60 | -33.66 | -33.21 | -33.08 | -33.18 | -29.36 | -37.93 | -39.00 | -37.86 | -39.22 | -35.50 |
02 Pink noise | -11.89 | -13.28 | -15.78 | -17.62 | -16.49 | -11.41 | -13.66 | -15.22 | -15.70 | -15.09 | -11.29 | -13.73 | -14.54 | -14.99 | -14.35 | -11.86 | -13.32 | -15.89 | -17.39 | -16.48 |
03 Electronic gong | -32.37 | -46.99 | -30.94 | -48.19 | -31.23 | -29.33 | -36.12 | -29.12 | -36.34 | -27.43 | -27.88 | -34.42 | -27.37 | -34.38 | -25.80 | -31.91 | -39.42 | -31.11 | -39.73 | -30.80 |
04 Electronic gong | -32.39 | -49.27 | -30.96 | -52.83 | -31.24 | -29.71 | -38.09 | -29.16 | -38.51 | -27.46 | -28.38 | -34.18 | -27.49 | -33.43 | -25.83 | -31.99 | -39.04 | -31.12 | -39.38 | -30.80 |
05 Electronic gong | -32.40 | -50.24 | -30.99 | -55.69 | -31.28 | -29.39 | -37.57 | -29.16 | -37.90 | -27.43 | -28.51 | -40.57 | -28.07 | -41.36 | -26.32 | -31.92 | -42.18 | -31.11 | -42.64 | -30.80 |
06 Electronic gong | -22.01 | -37.89 | -29.02 | -40.25 | -27.28 | -20.91 | -31.33 | -26.62 | -33.04 | -22.32 | -20.40 | -28.76 | -24.37 | -30.35 | -20.49 | -21.93 | -39.04 | -29.55 | -41.60 | -27.01 |
07 Electronic tune | -31.05 | -38.96 | -29.47 | -37.97 | -29.57 | -28.10 | -34.69 | -27.74 | -35.01 | -25.92 | -27.01 | -35.75 | -26.44 | -36.21 | -24.58 | -30.63 | -38.18 | -29.63 | -37.62 | -29.14 |
08 Violin | -20.62 | -23.86 | -26.96 | -25.10 | -26.90 | -19.81 | -23.48 | -25.47 | -24.07 | -23.78 | -19.37 | -22.77 | -24.21 | -23.33 | -22.49 | -20.52 | -23.80 | -27.07 | -25.13 | -26.65 |
09 Viola | -22.13 | -24.96 | -27.93 | -26.14 | -28.12 | -21.03 | -24.37 | -26.56 | -25.00 | -24.94 | -20.63 | -23.04 | -25.23 | -23.43 | -23.68 | -22.00 | -25.28 | -27.95 | -26.52 | -27.85 |
10 Violoncello | -23.11 | -25.82 | -27.85 | -29.25 | -28.82 | -22.94 | -26.57 | -27.62 | -26.76 | -25.87 | -22.66 | -25.27 | -26.82 | -25.57 | -25.03 | -23.40 | -26.89 | -27.82 | -28.92 | -28.02 |
11 Double-bass | -25.09 | -25.40 | -26.58 | -26.95 | -28.05 | -26.06 | -26.48 | -28.18 | -26.97 | -26.50 | -25.50 | -24.96 | -26.94 | -26.06 | -26.02 | -24.89 | -27.12 | -26.50 | -27.47 | -27.96 |
12 Picollo | -26.32 | -31.12 | -29.13 | -32.75 | -29.23 | -24.66 | -29.99 | -27.59 | -30.93 | -25.84 | -24.00 | -29.09 | -26.58 | -30.01 | -24.64 | -26.06 | -31.28 | -29.31 | -31.83 | -28.91 |
13 Flute | -25.36 | -30.03 | -29.46 | -31.85 | -29.44 | -23.80 | -28.86 | -28.13 | -29.93 | -25.86 | -23.21 | -27.95 | -26.58 | -28.65 | -24.41 | -25.13 | -29.73 | -29.73 | -31.47 | -29.10 |
14 Oboe | -21.48 | -26.94 | -25.89 | -29.09 | -26.29 | -19.86 | -26.13 | -24.26 | -28.09 | -22.96 | -19.10 | -24.26 | -23.41 | -25.08 | -21.92 | -21.29 | -26.96 | -26.03 | -29.85 | -25.91 |
15 Cor anglais | -23.08 | -27.31 | -27.50 | -29.28 | -27.78 | -21.88 | -26.75 | -26.01 | -27.70 | -24.39 | -21.26 | -25.50 | -24.77 | -26.25 | -23.15 | -22.95 | -27.20 | -27.50 | -28.66 | -27.33 |
16 Clarinet | -27.92 | -30.04 | -32.96 | -31.68 | -32.94 | -26.91 | -28.88 | -32.43 | -29.70 | -29.27 | -26.54 | -23.25 | -30.97 | -23.29 | -27.97 | -27.63 | -30.01 | -33.64 | -31.55 | -32.61 |
17 Bass-clarinet | -28.85 | -31.08 | -33.72 | -32.69 | -33.48 | -28.21 | -29.84 | -33.40 | -30.18 | -30.00 | -27.59 | -26.91 | -31.20 | -27.21 | -28.44 | -28.77 | -30.83 | -34.30 | -31.68 | -33.19 |
18 Bassoon | -22.63 | -23.07 | -26.83 | -25.59 | -27.05 | -23.30 | -24.52 | -26.90 | -25.16 | -25.68 | -23.63 | -24.67 | -26.48 | -25.56 | -25.16 | -22.72 | -23.20 | -26.81 | -25.44 | -26.73 |
19 Contra-bassoon | -14.02 | -15.97 | -17.23 | -19.18 | -17.96 | -13.93 | -17.71 | -17.74 | -18.76 | -17.44 | -13.99 | -16.78 | -17.11 | -17.55 | -17.29 | -14.11 | -16.70 | -17.08 | -19.94 | -17.94 |
20 Saxophone | -25.68 | -26.62 | -27.85 | -29.30 | -28.45 | -24.36 | -26.64 | -27.57 | -26.28 | -26.36 | -24.32 | -25.77 | -27.41 | -25.80 | -25.60 | -25.86 | -27.62 | -27.71 | -27.58 | -28.40 |
21 Trumpet | -25.38 | -30.40 | -28.94 | -32.94 | -29.22 | -23.98 | -28.91 | -27.64 | -30.20 | -25.98 | -23.16 | -26.76 | -26.43 | -27.81 | -24.67 | -25.31 | -30.16 | -29.01 | -32.45 | -28.90 |
22 Trombone | -29.11 | -31.35 | -31.03 | -33.43 | -31.45 | -28.05 | -30.19 | -30.77 | -30.54 | -28.27 | -27.72 | -27.37 | -29.85 | -27.59 | -27.37 | -28.78 | -31.44 | -31.15 | -32.46 | -30.92 |
23 Horn | -23.82 | -26.00 | -26.39 | -28.21 | -27.30 | -22.83 | -25.61 | -25.94 | -25.59 | -24.87 | -22.44 | -24.71 | -25.13 | -24.62 | -24.03 | -23.53 | -26.41 | -26.44 | -27.13 | -26.76 |
24 Tuba | -26.91 | -28.03 | -29.85 | -30.39 | -30.19 | -26.63 | -29.45 | -29.78 | -29.99 | -28.18 | -26.37 | -26.93 | -29.26 | -27.39 | -27.30 | -26.83 | -28.03 | -29.81 | -29.71 | -30.48 |
25 Harp | -18.82 | -20.76 | -21.04 | -23.43 | -21.81 | -20.01 | -23.50 | -21.30 | -24.27 | -21.97 | -20.17 | -23.57 | -21.63 | -24.34 | -21.18 | -19.32 | -21.71 | -20.21 | -23.33 | -21.96 |
26 Claves | -9.09 | -20.09 | -11.06 | -23.63 | -12.21 | -7.99 | -13.97 | -10.79 | -20.41 | -10.47 | -7.79 | -17.36 | -10.41 | -20.42 | -10.15 | -8.84 | -17.19 | -10.45 | -23.13 | -11.88 |
27 Castanets | -7.26 | -14.68 | -10.20 | -19.95 | -11.22 | -6.76 | -11.28 | -9.91 | -16.52 | -10.36 | -6.72 | -13.64 | -10.15 | -17.15 | -10.47 | -7.17 | -13.39 | -10.16 | -19.54 | -11.12 |
28 Side drum | -14.16 | -25.57 | -14.55 | -29.64 | -17.00 | -12.59 | -19.85 | -13.22 | -24.80 | -14.33 | -12.29 | -21.98 | -12.72 | -26.29 | -13.64 | -13.83 | -23.18 | -14.38 | -28.73 | -16.89 |
29 Bass drum | -12.71 | -15.90 | -16.37 | -21.25 | -17.74 | -13.02 | -17.71 | -14.07 | -19.69 | -15.50 | -13.29 | -16.47 | -14.65 | -16.71 | -15.00 | -12.75 | -17.02 | -15.66 | -21.49 | -16.38 |
30 Kettle-drums | -14.13 | -17.01 | -16.70 | -22.03 | -17.81 | -13.75 | -17.52 | -15.43 | -18.98 | -16.09 | -13.54 | -17.33 | -15.08 | -18.30 | -15.52 | -14.05 | -17.31 | -16.54 | -21.39 | -17.47 |
31 Cymbal | -13.20 | -16.02 | -16.22 | -20.25 | -17.13 | -12.60 | -15.77 | -15.09 | -17.65 | -15.49 | -12.19 | -15.75 | -15.17 | -16.79 | -14.73 | -13.23 | -15.77 | -15.97 | -19.86 | -17.01 |
32 Triangle | -23.55 | -35.08 | -24.97 | -37.67 | -26.25 | -21.98 | -14.71 | -23.48 | -13.94 | -23.57 | -21.66 | -21.41 | -22.97 | -21.17 | -22.74 | -23.33 | -30.74 | -24.68 | -33.67 | -25.98 |
33 Gong | -12.71 | -14.47 | -16.58 | -18.85 | -17.17 | -12.25 | -14.68 | -16.15 | -16.60 | -15.98 | -12.10 | -14.51 | -15.58 | -15.85 | -15.25 | -12.63 | -14.61 | -16.81 | -19.09 | -17.66 |
34 Tubular bells | -19.47 | -22.77 | -24.15 | -23.83 | -25.19 | -18.27 | -21.91 | -23.39 | -21.40 | -22.92 | -17.42 | -19.33 | -22.72 | -19.15 | -21.95 | -19.44 | -23.10 | -23.99 | -24.21 | -24.94 |
35 Glockenspiel | -21.02 | -33.07 | -22.74 | -36.13 | -24.09 | -19.89 | -25.98 | -21.53 | -31.76 | -21.77 | -19.71 | -28.68 | -20.89 | -31.66 | -21.11 | -20.90 | -28.84 | -22.46 | -34.27 | -23.79 |
36 Xylophone | -12.73 | -23.05 | -14.39 | -25.37 | -16.62 | -11.48 | -18.03 | -13.13 | -23.05 | -14.18 | -11.17 | -20.64 | -13.30 | -23.83 | -13.52 | -12.48 | -21.54 | -14.51 | -25.09 | -16.09 |
37 Vibraphone | -21.68 | -18.75 | -23.03 | -21.55 | -23.25 | -23.05 | -21.54 | -24.53 | -22.16 | -23.46 | -23.28 | -18.69 | -24.44 | -18.86 | -23.25 | -21.81 | -19.77 | -23.21 | -21.67 | -22.77 |
38 Marimba | -25.03 | -31.15 | -27.06 | -33.69 | -27.97 | -24.11 | -28.77 | -25.78 | -31.54 | -25.31 | -23.75 | -29.91 | -25.45 | -31.48 | -24.43 | -24.71 | -30.02 | -26.84 | -32.33 | -27.55 |
39 Grand piano | -21.14 | -23.48 | -23.51 | -26.48 | -23.98 | -20.20 | -23.96 | -22.82 | -24.51 | -22.53 | -19.94 | -23.16 | -22.11 | -23.97 | -21.60 | -20.84 | -24.73 | -23.42 | -25.82 | -23.99 |
40 Harpsichord | -16.96 | -20.77 | -21.55 | -22.06 | -22.18 | -16.59 | -19.42 | -20.60 | -20.94 | -20.13 | -16.44 | -19.92 | -19.62 | -20.24 | -19.35 | -16.94 | -20.25 | -21.40 | -21.78 | -22.06 |
41 Celesta | -25.90 | -27.41 | -26.71 | -30.55 | -27.42 | -25.08 | -28.91 | -26.57 | -29.71 | -26.11 | -24.59 | -27.04 | -25.51 | -27.55 | -25.08 | -25.87 | -29.26 | -28.22 | -30.74 | -28.19 |
42 Accordion | -19.19 | -22.34 | -24.67 | -24.25 | -24.93 | -18.25 | -21.27 | -23.37 | -23.15 | -21.91 | -17.85 | -21.10 | -22.20 | -22.57 | -20.89 | -18.92 | -22.15 | -24.76 | -24.21 | -24.67 |
43 Organ | -25.90 | -26.57 | -28.48 | -29.30 | -29.20 | -24.66 | -25.32 | -27.93 | -26.01 | -26.38 | -24.22 | -23.87 | -27.06 | -24.25 | -25.34 | -25.70 | -25.77 | -28.08 | -27.50 | -28.92 |
44 Soprano | -18.62 | -23.91 | -24.27 | -27.48 | -24.74 | -17.38 | -23.23 | -22.42 | -24.55 | -21.24 | -16.87 | -22.52 | -21.18 | -23.15 | -19.95 | -18.57 | -24.24 | -24.03 | -27.47 | -24.35 |
45 Alto | -17.15 | -21.78 | -22.53 | -25.08 | -23.02 | -15.85 | -21.34 | -20.81 | -22.90 | -20.02 | -15.16 | -20.76 | -19.81 | -21.29 | -18.90 | -16.96 | -22.06 | -22.80 | -25.06 | -22.86 |
46 Tenor | -18.54 | -23.01 | -23.52 | -26.26 | -24.03 | -17.27 | -22.30 | -21.71 | -23.89 | -20.84 | -16.73 | -21.67 | -20.56 | -22.53 | -19.59 | -18.41 | -23.24 | -23.51 | -26.34 | -23.76 |
47 Bass | -15.36 | -18.16 | -20.37 | -21.69 | -20.96 | -14.48 | -18.42 | -18.69 | -19.95 | -18.17 | -14.05 | -17.59 | -17.73 | -18.63 | -17.15 | -15.29 | -18.34 | -20.39 | -21.47 | -20.73 |
48 Quarter | -15.76 | -17.69 | -20.81 | -21.45 | -21.60 | -15.15 | -18.68 | -19.69 | -19.81 | -19.22 | -15.05 | -18.28 | -18.74 | -19.09 | -18.22 | -15.80 | -18.28 | -20.81 | -21.23 | -21.59 |
49 Female speech eng. | -15.66 | -22.26 | -18.30 | -26.17 | -19.44 | -14.44 | -21.33 | -16.53 | -24.06 | -16.75 | -13.99 | -21.85 | -15.75 | -23.78 | -15.80 | -15.48 | -21.91 | -18.21 | -25.87 | -19.19 |
50 male speech eng. | -16.48 | -23.80 | -19.41 | -26.66 | -20.36 | -15.04 | -22.12 | -17.01 | -23.79 | -17.22 | -14.48 | -21.62 | -15.96 | -24.33 | -16.11 | -16.28 | -23.32 | -19.23 | -25.96 | -20.03 |
51 Female speech fra. | -18.81 | -26.02 | -20.99 | -30.17 | -22.09 | -17.36 | -24.64 | -19.08 | -27.22 | -19.23 | -16.88 | -24.76 | -18.37 | -26.98 | -18.31 | -18.67 | -25.61 | -20.69 | -29.65 | -21.83 |
52 male speech fra. | -16.73 | -22.14 | -19.35 | -25.15 | -20.22 | -15.66 | -21.60 | -17.75 | -23.66 | -17.62 | -15.11 | -22.06 | -16.96 | -23.46 | -16.79 | -16.56 | -21.80 | -19.22 | -25.27 | -19.81 |
53 Female speech ger. | -15.94 | -21.36 | -19.33 | -22.91 | -20.44 | -14.81 | -19.75 | -17.73 | -24.06 | -17.66 | -14.23 | -19.67 | -16.99 | -23.36 | -16.90 | -15.63 | -20.97 | -19.19 | -22.22 | -20.14 |
54 Male speech ger. | -17.30 | -23.02 | -20.18 | -26.86 | -21.07 | -15.86 | -22.38 | -18.10 | -24.82 | -18.08 | -15.28 | -22.31 | -17.24 | -23.89 | -17.07 | -17.10 | -22.91 | -19.99 | -26.37 | -20.85 |
55 Trumpet-Haydn | -17.70 | -19.29 | -20.91 | -21.17 | -21.56 | -16.09 | -19.85 | -19.79 | -20.58 | -19.49 | -15.87 | -18.88 | -19.17 | -19.47 | -18.60 | -17.57 | -20.77 | -20.82 | -21.78 | -21.28 |
56 Organ-Handel | -16.37 | -16.84 | -20.14 | -19.67 | -20.97 | -15.98 | -16.97 | -19.64 | -17.47 | -19.65 | -15.82 | -16.60 | -19.60 | -16.79 | -19.25 | -16.19 | -17.04 | -20.14 | -19.45 | -21.08 |
57 Organ-Bach | -12.49 | -17.81 | -18.00 | -20.66 | -18.99 | -10.69 | -17.40 | -17.30 | -17.41 | -17.16 | -10.17 | -16.44 | -16.34 | -15.62 | -16.36 | -12.52 | -17.86 | -17.99 | -20.66 | -19.08 |
58 Guitar-Sarasate | -18.41 | -20.17 | -20.75 | -23.80 | -21.82 | -17.77 | -20.60 | -20.34 | -21.30 | -20.07 | -17.50 | -20.98 | -20.13 | -21.34 | -19.99 | -18.40 | -20.75 | -20.55 | -23.10 | -21.43 |
59 Violin-Ravel | -18.68 | -21.37 | -24.18 | -23.13 | -24.54 | -17.69 | -21.25 | -22.54 | -22.05 | -21.66 | -17.33 | -20.59 | -21.69 | -21.72 | -20.58 | -18.58 | -21.55 | -23.98 | -23.04 | -24.23 |
60 Piano-Schubert | -16.59 | -18.86 | -20.67 | -22.67 | -21.52 | -16.05 | -19.53 | -20.25 | -20.15 | -19.77 | -15.84 | -18.68 | -19.39 | -19.02 | -19.04 | -16.56 | -19.37 | -20.62 | -22.23 | -21.51 |
61 Soprano-Mozart | -15.30 | -18.88 | -19.11 | -21.79 | -20.16 | -14.34 | -18.86 | -17.76 | -19.98 | -17.44 | -14.00 | -18.02 | -16.80 | -18.88 | -16.47 | -15.23 | -19.23 | -18.95 | -21.71 | -19.96 |
62 Soprano-Spiritual | -17.17 | -23.07 | -21.78 | -25.87 | -22.59 | -16.02 | -21.83 | -19.18 | -22.79 | -18.77 | -15.53 | -21.37 | -17.80 | -22.60 | -17.37 | -17.03 | -23.21 | -21.70 | -24.96 | -22.24 |
63 Soloists-Verdi | -12.91 | -17.17 | -16.43 | -21.26 | -17.36 | -12.01 | -16.87 | -14.73 | -19.15 | -14.88 | -11.74 | -16.70 | -13.94 | -18.20 | -14.09 | -12.75 | -17.24 | -16.24 | -21.04 | -17.17 |
64 Choir-Orff | -14.13 | -16.32 | -18.10 | -19.83 | -19.11 | -13.34 | -16.79 | -17.45 | -18.19 | -17.20 | -13.06 | -16.45 | -16.73 | -17.36 | -16.34 | -14.10 | -16.62 | -18.11 | -20.01 | -19.06 |
65 Orchestra-Strauss | -12.68 | -13.99 | -15.16 | -14.87 | -17.86 | -16.06 | -16.64 | -18.08 | -18.27 | -17.16 | -17.15 | -18.66 | -18.77 | -19.17 | -18.42 | -13.30 | -14.35 | -15.49 | -14.76 | -17.41 |
66 Wind ensemble-Stravinsky | -14.28 | -15.65 | -18.13 | -18.99 | -19.22 | -13.72 | -16.01 | -17.37 | -17.12 | -17.28 | -13.53 | -15.72 | -16.94 | -16.53 | -16.52 | -14.22 | -15.93 | -18.17 | -19.15 | -19.06 |
67 Wind ensemble-Mozart | -20.25 | -24.91 | -23.57 | -26.96 | -24.44 | -18.71 | -23.97 | -21.78 | -24.74 | -21.39 | -18.04 | -23.39 | -21.43 | -23.54 | -20.70 | -20.13 | -24.55 | -23.34 | -26.11 | -24.00 |
68 Orchestra-Baird | -14.00 | -15.98 | -17.86 | -19.47 | -18.82 | -13.27 | -15.88 | -16.93 | -17.08 | -16.80 | -13.37 | -15.73 | -16.62 | -16.55 | -16.47 | -13.87 | -16.38 | -17.69 | -19.63 | -18.80 |
69 ABBA | -14.02 | -19.81 | -16.05 | -22.76 | -17.20 | -13.06 | -18.57 | -15.37 | -21.28 | -15.63 | -12.86 | -18.37 | -15.00 | -19.60 | -15.41 | -13.92 | -19.88 | -15.84 | -22.90 | -17.07 |
70 Eddie Rabbitt | -16.08 | -18.17 | -18.72 | -21.76 | -19.76 | -15.31 | -18.99 | -17.99 | -20.22 | -17.80 | -15.04 | -19.01 | -17.74 | -19.84 | -17.19 | -16.17 | -18.66 | -18.70 | -22.02 | -19.53 |
Avg. | -17.88 | -21.79 | -21.13 | -24.87 | -22.07 | -17.09 | -21.11 | -20.19 | -22.74 | -19.88 | -16.79 | -20.99 | -19.59 | -21.98 | -19.13 | -17.79 | -21.81 | -21.04 | -24.61 | -21.85 |