A Spectral Envelope Estimation Method Based on F0-Adaptive Multi-Frame
Integration Analysis
Tomoyasu Nakano and Masataka Goto
National Institute of Advanced Industrial Science and Technology (AIST), Japan{t.nakano, m.goto}[at]aist.go.jp

Abstract
This paper presents a novel method of spectral envelope estimation and representation. Despite much sophisticated workin this area, estimating an appropriate envelope is still difficult.
We therefore propose an F0 -adaptive multi-frame integration
analysis method for estimating spectral envelopes with appropriate shape and high temporal resolution. The method does
not use pitch marks or phoneme labels and can be used withvarious types of sound (speech, singing, and instruments). The
basic idea is to use F0 -adaptive window analysis with a small
window length yielding high temporal resolution. The analysis is then extended by using neighboring frames to obtain astable spectral envelope. In tests using synthesized sound andresynthesized natural sound samples, for 8 of 14 samples thelog-spectral distances obtained with the proposed method were
smaller than those obtained with well-known previous methods.
Index Terms: spectral envelope, periodic signals, source-filter
model, speech/singing, instrument sound1. Introduction
Source-filter processing [1] is an important way to deal withspeech, singing, and instrument sounds. An appropriate spectral envelope reconstructed from an observed spectrum canbe useful in flexible sound manipulation (synthesis) and insound recognition (analysis). The aim of the work reported
here was to develop a method for estimating a spectral envelope with appropriate shape and high temporal resolution for
high-quality sound synthesis and high-accuracy sound analysis and to do so without using pitch marks1 and phoneme labels so that the method can be used with any kind of sound(speech/singing/instrument).
Various sophisticated methods for estimating spectral envelopes have been proposed. One of the best known signal
modeling techniques is that used in the phase vocoder [2], andmany extensions of it have been proposed (e.g., [3]). This technique successfully models and synthesizes harmonic signals
with static fundamental frequency (F0 ) characteristics but has
limitations with regard to the synthesis of sounds with changing
F0 . Although PSOLA [1, 4] is a well known signal manipulating method that can be used with F0 -changing sounds, it needs
precise pitch marks.
Some other widely known spectral envelope representation
methods use linear predictive coding (LPC) [5, 6], line spectral pairs (LSP), or cepstrum. LPC has been extended to melgeneralized cepstral analysis for treating various spectral representations [7], and the cepstrum-based methods have also been
improved [8–10] and combined with the LPC method [11]. Although the envelopes estimated using these methods are better(with regard to spectral shape) than those estimated using the
1 Time instants for F -synchronous analysis [1].
0simple cepstrum and LPC methods, the detail of the envelope(e.g., the sharpness of its peaks and valleys) is limited by theorder of the cepstral/LPC approximation.
The most well known alternatives are the sinusoidal models [12, 13], for which there have been many extensions – suchas making distinctions between the sinusoidal and broadband
spectrum components or noise in the signal [14, 15], extracting
the sinusoids from the spectrogram [16], extracting sinusoidsin an iterative way [17, 18], using quadratic interpolation [19],getting higher temporal resolution [20], monaural speech separation [21], and applying sinusoidal models to nonstationary
sounds [22, 23]. These sinusoidal approaches can be used forhigh-quality sound synthesis, and some have high temporal resolution [20, 22]. A sinusoidal model of the harmonic structure,
however, has energy only at frequencies corresponding to integer multiples of F0 . It is therefore difficult to identify transfer
characteristics between adjacent harmonics.
STRAIGHT [24] is a well known high-quality sourcefilter processing system (i.e., vocoder) and is widely usedin the speech research community.
From input speechsound it can estimate an interference-free spectrum basedon F0 -adaptive analysis. An extension/reformulation called
TANDEM-STRAIGHT [25] is based on calculating a temporally stable power spectrum, and either STRAIGHT or
TANDEM-STRAIGHT can be used to estimate the envelopeof an aperiodic component. The STRAIGHT spectral envelope can be used for high-quality sound synthesis and highaccuracy sound analysis with high temporal resolution without
pitch marks and phoneme labels. Although this spectral envelope can be quite accurate, it is obtained by spectral smoothing.
This means that even with the STRAIGHT spectral envelope it
is difficult to identify transfer characteristics between adjacentharmonics.
This is also true when the spectral envelope is estimated using Gaussian mixture modeling, such as STRAIGHT envelopemodeling [26], and when a Gaussian mixture function is usedin the joint estimation of the F0 and the spectral envelope [27].
Several methods for statistically estimating the envelope
from multiple frames have been studied in efforts to deal withthis problem [28–30], and this multi-frame approach has also
been used in a method estimating spectral envelopes of vocal
from music sound mixtures [31]. This multi-frame integration
approach assumes that additional information, such as phoneme
transcriptions, is available for selecting frames at which thespectral envelopes are presumed to be similar to each other.
In this approach not only the harmonic components observed
at each analyzed frame but also those at other frames are considered so that missing frequency parts can be interpolated statistically. This kind of multi-frame analysis, however, needs
phoneme transcriptions and has difficulties dealing with soundvarieties and context dependencies.

Overlapped windowed waveforms

A periodic signal (speech/singing/instrument sound)tt - 1/(2*F0)

t + 1/(2*F0)

time

F0-adapve
Gaussian windows(The standard deviaonis set to 2/F0)
amplitude [dB]

40
20
0

Valleysgroup delay [s]amplitude
amplitude

F0 = 318.6284 Hz

Windowing

time

FFT

Spectrum0.0499time [s]60-20Waveform

0.04640.0408

0.1

0.04
0
0

1

2

3

4

frequency [kHz]Figure 2: Overlapped windowed waveforms for F0 -adaptive
Gaussian windows (top). Corresponding STFT results (middle)
and group delays (bottom).
frequency

Integraon

Amplitude

Spectral envelope

frequency

Figure 1: Overview of F0 -adaptive multi-frame integration
analysis.

This paper presents a novel method of spectral envelope estimation and representation: F0 -adaptive multi-frame integration analysis. The basic idea is to use F0 -adaptive analysis witha small window length yielding high temporal resolution. The
analysis is extended to a multi-frame integration approach thatuses only neighborhood frames. This integration analysis does
not need phoneme transcriptions, can estimate stable spectralenvelopes without using precise pitch marks, and can determine
the amplitude range for each frequency bin of the spectral envelope. Like the other methods, our method has difficulty identifying transfer characteristics between adjacent harmonics but
this amplitude range information provides a better estimate of
the spectral envelope.

2. F0 -adaptive multi-frame integration
analysis
Figure 1 shows an overview of the proposed F0 -adaptive multiframe integration analysis. First, F0 -adaptive Gaussian windows are used for spectrum analysis (F0 -adaptive analysis).
Second, neighborhood frames are integrated to estimate the target spectral envelope (multi-frame integration analysis).
Overlapped windowed waveforms in which F0 -adaptive
Gaussian windows are shifted at each wave sample are shownin Figure 2 along with their corresponding short-term Fouriertransform (STFT) results (overlapped spectra) and their groupdelays2 . This figure suggests that we could obtain stable spectral envelopes by integrating these STFT results. When a single
window without any integration is used, it often suffers fromproblematic “valleys” in the spectrum as shown in the middleof Figure 2. These valleys are interference effects caused by
the distance between the center of a Gaussian window and theposition of the periodically localized energy of the audio signal.
Note that these valleys also correspond to positive and negative
peaks in the group delays as shown in the bottom of Figure 2,
where each peak means that the localized energy is far from thecenter of the window.
Our F0 -adaptive multi-frame integration analysis therefore
integrates the STFT results (overlapped spectra) using multiple
Gaussian windows to fill in these valleys. As shown in Figure
3, we can obtain the maximum and minimum envelopes fromthe overlapped spectra. We call the areas between the maximum and minimum envelopes amplitude ranges. This amplitude range information is an informative representation because
the actual spectral envelope without any problematic valleys
should always stay in this area. Our method finally estimates
the target spectra envelope by averaging the maximum and minimum envelopes so that the estimated envelope can be in thisarea.
2.1. Conditions for implementation
To estimate a spectral envelope by F0 -adaptive analysis, we assume that the F0 of a signal has already been estimated by any
appropriate technique. Throughout this paper, sound samples
are monaural recordings without accompaniment or noise andare digitized at 16 bit / 44.1 kHz.

2 The group delay is computed by using a method described in [32].

Waveform

Waveform

N

0

o

m

e

Spectrogram

22.05

0

0.1

0.2

0.3

0.4

0.5

0.6

n

o

m

e

PhonemeF0-adaptive spectrogram

0time [s]Proposed envelope
Amplitude ranges

0

2.2

frequency [kHz]-20
amplitude [dB]N

Phonemefrequency [kHz]frequency [kHz]

22.05

n

-40

-60

0
Temporal contour (at 645.9941 Hz)

-80
0

5

10

15

frequency [kHz]Figure 3: A waveform of singing (top) and its spectral envelope
and amplitude ranges estimated by F0 -adaptive multi-frame integration analysis (middle and bottom).

F0-related ﬂuctuation

Figure 4: A waveform and its F0 -adaptive spectrogram without pitch marks (top). A close-up view of the waveform and thespectrogram (middle), and a temporal contour of the spectrogram at 645.9961 Hz (bottom).

2.2. F0 -adaptive analysis
We designed an F0 -adaptive window by using a Gaussian function (Figure 1). Let w(τ ) be a Gaussian window function oftime τ defined as follows, where σ(t) is the standard deviationof the Gaussian distribution and F0 (t) is the fundamental frequency for analysis time t. The window is normalized by theroot mean square (RMS) value calculated with N defined as the
FFT length.
ŵ(τ )∑ −1
2(1/N ) N
τ =0 ŵ(τ )

w(τ )

=

√

ŵ(τ )

=

exp(−

σ(t)

=

τ2
)
2σ(t)2
1
1
×F0 (t)
3

(1)

(2)
(3)

The Gaussian window’s σ(t) = 1/(3 × F0 (t)) means the analysis window length (2 × 3σ) is set to two fundamental periods(2 × 3σ = 2/F0 (t), see Figure 1). This length is known to givea good approximation of the local spectral envelope [1]. The
results of F0 -adaptive window analysis have F0 -related fluctuation along the time axis as shown in the bottom part of Figure
4.
Since the Gaussian windows are shifted at each wave sample, the discrete time-shift of the F0 -adaptive analysis is set to
one wave sample (1/44100 s).
2.3. Multi-frame integration analysis
This multi-frame integration analysis first overlaps neighborhood spectra as shown in Figure 2. The range of the overlappingis from −1/(2 × F0 ) to 1/(2 × F0 ), which can cover a periodof harmonic signals (Figure 1). The results of our preliminary
experiment showed that a smaller range is not sufficient to fillin the spectral valleys.
By integrating the neighborhood spectra, one obtains the
maximum and minimum envelopes of those spectra as the upperand lower bounds of the target spectral envelope. Our method
then estimates the target envelope by averaging the maximum
and minimum envelopes. Before this averaging, however, thevalleys in the minimum envelope should be eliminated because
frequency components around the valleys are not appropriately
observed. For this elimination, we transform the maximum envelope into the new valley-free minimum envelope that touches
positive peaks of the old (original) minimum envelope with valleys as shown in Figure 5. After the ratio of this transformation
at each peak (at its frequency bin) is set, the ratio of the transformation at the other frequency bins is linearly interpolated andadjusted so that the transformed envelope can be higher than theold minimum envelope.
Before the maximum and new minimum envelopes are averaged, they are smoothed along both the time and frequency
axes by using an 2-dimensional FIR low-pass filter. As shownin Figure 6, for example, the temporal contour of the maximum
envelope is much smoother than the F0-adaptive spectrum before the integration, but still has a step-like contour that can befurther smoothed by the low-pass filter.
Since the estimated envelope of frequency bins under F0 is
known to be unreliable, we finally smooth the envelope of thosebins so that it can be same with the value of the envelope at F0
as shown in Figure 5.

(3) The rates are linearly interpolated ( )(2) Determine conversion rates ( )(4) Transformation ( )Higher than the minimum envelope ( )(5) Averaging, andreplacing values under F0 ( )

0
22.05

Spectrogram (STRAIGHT)

frequency [kHz](1) Peak ( ) picking

Spectrogram (proposed)

frequency [kHz]22.05

Maximum envelope
Minimum envelope
Peaks of the minimum envelope
New minimum envelope

00transformed from the maximum envelope

0.1

0.2

0.3

0.5

0.6

time [s]Estimated spectral envelope

F0

amplitude [dB]40

STRAIGHT envelope
Maximum envelope
Minimum envelope

20
0
-20

Proposed envelope

-40
-60
1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 frequency [kHz]Figure 5: The target spectral envelope (green line) is estimatedby averaging the maximum envelope (blue line) and the newminimum envelope (black line).

60
40
amplitude [dB]

0

20
0
-20

F0-adaptive spectrum
Maximum envelope
Maximum envelope (smoothed)

-40
0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 frequency [kHz]Figure 7: Comparison of the spectrogram estimated by the proposed method (top) and the STRAIGHT spectrogram (middle).
The corresponding spectral envelopes at time 0.4 sec can also
be compared (bottom).
time

Figure 6: Temporal contours of the maximum envelope before and after the low-pass filter. Integration smoothes the F0 adaptive spectrum (blue line, same as that in the bottom of Figure 4) to the maximum envelope (red line), which still has steps
that can be eliminated (green line).

3. Experimental evaluations
In two experiments the proposed method was comparedwith two previous methods, STRAIGHT [24] and TANDEMSTRAIGHT [25]. An unaccompanied male singing sound (solovocal) was taken from the RWC Music Database3 , a female
spoken sound was taken from the AIST Humming Database(E008) [34], and two kinds of instrument sounds were takenfrom the RWC Music Database4 .
All spectral envelopes were represented by 2049 frequency
bins (4096 FFT length) and had a temporal resolution of 1 ms,
which was the discrete time step in the analyses.

3 Music Genre [33]: RWC-MDB-G-2001 No. 91.
4 Musical Instrument Sound [33]: RWC-MDB-I-2001 No.
01
011PFNOM as a piano sound, and No.16 161VLGLM as a violin sound.

3.1. Experiment A: comparison
By using natural sound samples, we compared spectral envelopes estimated by the proposed method with spectral envelopes estimated by STRAIGHT [24], which are known to behighly accurate.
In Figure 7 the frequency spectrogram obtained using
our proposed method is shown along with the corresponding
STRAIGHT spectrogram. The spectral envelope obtained using STRAIGHT lies between our maximum and minimum envelopes and is similar to the proposed envelope. We also foundthat sounds resynthesized using the spectrogram obtained with
our method can give an impression of naturalness comparable tothat given by sounds resynthesized using the STRAIGHT spectrogram.
3.2. Experiment B: simulation
We evaluated the accuracy of the spectral envelope estimationby comparing the proposed method with STRAIGHT [24] andTANDEM-STRAIGHT [25]. To prepare the ground truth forevaluating its accuracy, we used sounds resynthesized from natural sound samples by using STRAIGHT and also used sounds
synthesized by a Klatt synthesizer.
For sounds resynthesized by using STRAIGHT, we first
analyzed the natural sound samples (singing, speech, and instrument sounds) to estimate the F0 and the STRAIGHT spec-Table 1: Control parameters for cascade-type Klatt synthesizer[35] in experiment B.
Name
Fundamental frequency
First formant frequency
Second formant frequency
Third formant frequency
Fourth formant frequency
Fifth formant frequency
First formant bandwidth
Second formant bandwidth
Third formant bandwidth
Fourth formant bandwidth
Fifth formant bandwidth
Glottal resonator frequency
Glottal resonator bandwidth

Value (Hz)125
250–1250
750–2250
2500
3500
4500
62.5
62.5
125
125
125
0
100

Proposed envelope
Amplitude ranges(Maximum & Minimum envelope)-40

F1 (Hz)
250
250
500

F2 (Hz)
750
1500
1500

Sample No.
K04
K05
K06

F1 (Hz)
1000
1000
500

-60
-80
-100

Table 2: F1 and F2 values for cascade-type Klatt synthesizer[35] in experiment B.
Sample No.
K01
K02
K03

STRAIGHT
TANDEM-STRAIGHT

amplitude [dB]

Symbol
F0
F1
F2
F3
F4
F5
B1
B2
B3
B4
B5
FGP
BGP

Ground-truth

F2 (Hz)
1500
2000
2000

0

1

2

3

frequency [kHz]Figure 8: Comparison of Klatt (K05) spectral envelopes estimated by the proposed method and two previous methods.

4. Conclusion
trogram. We then resynthesized sound samples from the
STRAIGHT spectrogram with different F0 s. This STRAIGHT
spectrogram can be used as the ground truth for these resynthesized sound samples.
We synthesized six voice-like sounds by using a cascadetype Klatt synthesizer [35] with the parameters listed in Table 1.
F1 and F2 for the six sounds were set as shown in Table 2. Notethat we just obtained spectral envelopes from the Klatt synthesizer for the ground truth. We then used our own sinusoidal
synthesis implementation using the obtained spectral envelopes
to generate sound samples having the F0 of 125 Hz.
The spectral envelopes (spectrogram) estimated by the threemethods were evaluated in terms of the log-spectral distance
LSD defined by
LSD

=

α(t)

=

ϵ2

=

FH
T −1
Sg (t, f )1 ∑ 1 ∑
20 log10
T t=0 Fα(t) · Se (t, f )
f =FL
∑FH
f =FL Sg (t, f )Se (t, f )∑FH
2
f =FL Se (t, f )FH∑(Sg (t, f ) − α(t)Se (t, f ))

(4)

(5)

(6)

This paper proposed a simple, highly accurate, and flexible
spectral envelope estimation method called F0 -adaptive multiframe integration analysis. The method was evaluated by using synthesized/resynthesized sound samples and comparing its
results with those of two well-known methods for spectral envelope estimation. The proposed method can estimate accurate spectral envelopes with amplitude range information usefulfor flexible representation. The evaluation results suggest thatenvelopes estimated by the proposed method can be used forsound synthesis/analysis.
In future work we expect to improve the envelope estimation method, to investigate a method for estimating aperiodicity,phase information, and excitation, and to incorporate statisticaltechniques and additional information (e.g., phoneme transcriptions).

5. Acknowledgements
This research utilized the RWC Music Database “RWC-MDBG-2001” (Music Genre), “RWC-MDB-I-2001” (Musical Instrument Sound), and AIST Humming Database (E008). This research was supported in part by OngaCREST project (CREST,
JST).

f =FL

where T is the number of voiced frames, F is the number of
frequency bins (= FH − FL + 1), (FL , FH ) is the frequency
range for the evaluation, Sg (t, f ) and Se (t, f ) are respectively
the ground-truth and estimated spectral envelopes, and α(t) is anormalization factor determined by minimizing an error defined
as a square error ϵ2 between Sg (t, f ) and α(t)Se (t, f ) at eachframe t. The log-spectral distances obtained in this experiment
are listed in Table 3, and an estimated result is shown in Figure
8.
The envelope estimation accuracies obtained using our proposed method are sometimes better than those obtained using
STRAIGHT and TANDEM-STRAIGHT. This suggests that our
method can be used for high-quality sound synthesis and highaccuracy sound analysis.

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Table 3: Log-spectral distances for spectrograms obtained in experiment B when using two previous methods and the proposed method.
The smallest values are underlined, and the second smallest values are indicated by bold-faced type.
sound
type
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[kHz]
6
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5

LSD (log-spectral distance) [dB]
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3.12324.00243.59513.3649
1.14671.7994
1.37942.27112.36892.10122.3131
1.6676
1.94913.8462
1.59952.82781.67641.47002.29541.7053
1.26991.1271
1.57591.23531.0643
1.1712
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