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IEEE SIGNAL PROCESSING MAGAZINE [59] MARCH 2015

faster phase estimation from magnitude, modeling of the signal

phase, group delay and transient processing, and joint estimation

of phase and magnitude.

ITERATIVE ALGORITHMS FOR PHASE ESTIMATION

Among the first proposals for phase estimation are iterative

approaches, which aim at estimating a time-domain signal

whose STFT magnitude is as close as possible to a target one

[1], [8]. Indeed, if the STFT magni-

tude of two signals are close, the

signals will in general be perceptu-

ally close as well. Thus, finding a

signal whose STFT magnitude is

close to a target one is considered a

valid goal when looking to obtain a

signal that “sounds” like that target

magnitude. This motivated intense

research on algorithms to estimate

signals (or equivalently a corresponding phase) given target

magnitudes, with applications such as speech enhancement or

timescale modification. In the case of speech enhancement, the

magnitude is typically obtained through one of the many mag-

nitude estimation algorithms mentioned earlier, while some

estimate of the phase, such as that of the noisy mixture, may

further be exploited for initialization or as side information.

The most well known and fundamental of these approaches

is that of Griffin and Lim [1], which consists in applying STFT

synthesis and analysis iteratively while retaining information

about the updated phases and replacing the updated magni-

tudes by the given ones. This exploits correlations between

neighboring STFT frames to lead to an estimate of the spectral

phases and the time-domain signal.

Given a target magnitude spectrogram

,A Griffin and Lim

formulated the problem as that of estimating a real-valued time-

domain signal

x such that the magnitude of its STFT X is closest

to A in the least-squares sense, i.e., estimating a signal x which

minimizes the squared distance

(, ) || | | .xAdXA

(2)

They proposed an iterative procedure which can be proven to min-

imize, at least locally, this distance. Starting from an initial signal

estimate x

()0

such as random noise, iterate the following compu-

tations: compute the STFT X

()i

of the signal estimate x

()i

at step

;i compute the phase estimate

()i

z as the phase of ,X

()i

() ()ii

+z = compute the signal estimate x

()i 1+

at step i 1+ as

the iSTFT of .Ae

()i

Using the operator G defined in (1), this can

be reformulated as

().AeG

()i 1j

()i

+z =

(3)

This procedure can be proven to be nonincreasing as well for a

measure of inconsistency of the spectrogram Ae

()i

defined

directly in the time-frequency domain:

() ( ) .AAeeIG

z =-

(4)

Indeed, one can easily show that ( , ) ( ) ( , ) .xA xAddI

( ) () ()iii1

##z

Interestingly, if only parts of the phase are updated according to (3),

the nondecreasing property still holds for

(),I

z but whether it still

does for (, )xAd has not been established.

Due to the extreme simplicity of its implementation and to its

perceptually relatively good results, GL was used as the standard

benchmark and a starting point for multiple extensions in the

three decades that have followed, even after better and only mar-

ginally more involved algorithms

had been devised. Most of the algo-

rithms that have been developed

since attempted to fix GL’s issues, of

which there are several: first, conver-

gence typically requires many itera-

tions; second, GL does not provide a

good initial estimate, starting from

random phases with no considera-

tions for cross-frame dependencies;

third, the updates rely on computing STFTs, which are computa-

tionally costly even when implemented using fast Fourier trans-

forms (FFTs); fourth, the updates are typically performed on whole

frames, without emphasis on local regularities; and finally, the

original version of GL processes signals in batch mode.

On this last point, it is interesting to note that Griffin and Lim

did actually hint at how to modify their algorithm to use it for

online applications. They described briefly in [1] and with more

details in [16] how to sequentially update the phase using “cas-

caded processors” that each take care of one iteration; their partic-

ular proposal however still incurs an algorithmic delay of

I times

the window length if performing I iterations. In [16], Griffin also

presented several methods that he referred to as “sequential esti-

mation methods”: these only incur a single frame delay and could

thus be used for online application, the best performing one being

reported as on par with batch GL.

While one can already see in Griffin’s account [16] several ele-

ments to modify GL into an algorithm that can lead to high qual-

ity reconstruction in a real-time setting, such as sliding-block

analysis across the signal and the use of windows that compensate

for partially reconstructed frames, these ideas seem to have gone

largely unnoticed and it is not until much later that they were

incorporated into more refined methods. Beauregard, Zhu, and

Wyse proposed consecutively two algorithms for real-time signal

reconstruction from STFT magnitude, the real-time iterative spec-

trogram inversion (RTISI) algorithm and RTISI with look ahead

(RTISI-LA) [17]. RTISI aims at improving the original batch GL in

two respects: allowing for online implementation, and generating

better initial phase estimates. The algorithm considers the frames

sequentially in order, and at frame

,, it only uses information

from the current frame’s magnitude and the previous overlapping

frames. The initial phase estimate

()0

for frame , is obtained as

the phase of the partial reconstruction from the previous frames,

windowed by an analysis window, which already ensures some

consistency between the phases of the current and previous

frames. An iterative procedure similar to GL is then applied, lim-

ited to the current frame’s phase: at each iteration, frame

, ’s

FINDING A SIGNAL WHOSE

STFT MAGNITUDE IS CLOSE TO

A TARGET ONE IS CONSIDERED A

VALID GOAL WHEN LOOKING TO

OBTAIN A SIGNAL THAT “SOUNDS”

LIKE THAT TARGET MAGNITUDE.

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