Abstract:

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 However, the phase might be lost or no longer applicable depending on the kind of processing involved.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.

Additional content:

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
Trunc. GaussHannHammingBlackman
SPSIRTPGHI(0)RTISI(0)RTPGHI(1)RTISI(1)SPSIRTPGHI(0)RTISI(0)RTPGHI(1)RTISI(1)SPSIRTPGHI(0)RTISI(0)RTPGHI(1)RTISI(1)SPSIRTPGHI(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