IEEE Signal Processing - March 2018 - 164

~
x p (n )

~
x (n )

n

n
(b)

(a)

1 ~
[x (n ) + x~p∗ (-n )]
2 p

~
x p∗ (-n )

n

n
(d)

(c)

Figure 2. Plots showing the reconstruction of a discrete-time signal (with delay and zero-padding) from only the RDFT: (a) an extended real-valued
signal xu (n) with delay and zero padding (M = 2N + 1 = 9); (b) a periodic version of xu (n) (with a period of M ) as a result of the cyclic properties of the
DFT; (c) a complex conjugate and time-reversed version of xu p (n); and (d) an output signal from taking the inverse DFT of the real part, showing no timedomain aliasing. The shaded region contains the reconstructed signal.

phase in the synthesis stage. In particular, the enhanced speech from the application of minimum mean squared error
estimation methods in the modulationRI domain was found to have improved
speech intelligibility [6]. In the RI
approach to speech enhancement, the
enhancement algorithm is applied
independently on the RI parts, treating them as if they were time-domain
signals. The theoretical advantages of
modulation-RI processing include a
valid additive-noise assumption in this
domain as well as eliminating the step
of combining with the noisy DFT phase
in the synthesis stage [5].
First, there can be computational
advantages of the RDFT method of signal reconstruction in the analysis-modification-synthesis (AMS) framework
[7] that is used in the modulation-RI
enhancement method, because the latter
method requires the processing of two
sets of signals (R and I) as opposed to
the former method (R only). In the AMS
framework, speech is windowed into
overlapping short (e.g., 32 ms) frames
and then a densely sampled DFT is computed for each frame. For speech signals
sampled at 8 kHz, each frame would
consist of 256 time-domain samples. In
the case of a critically sampled M-point
complex-valued DFT (where M = 256),
164

the computational complexity of the
independent processing of RI parts
would be identical to the RDFT (i.e.,
257 real numbers), which itself requires
a 513-point DFT . More specifically for
this particular case, a 256-point complex-valued DFT would produce 256
real and 256 imaginary values. Since a
real-valued signal is being considered,
the complex-conjugation property of the
DFT means that only 129 real (including
the dc value) and 128 imaginary values
need to be processed, which gives a total
of 257.
For the RDFT processing, zero-padding is applied and a 513-point complex-valued DFT would produce 257
unique real values. However, a more
densely sampled DFT is typically used
in modulation-domain speech processing because of its finer spectral interpolation properties. In the densely sampled
DFT case (e.g., M = 512), the RI processing requires 513 unique values to
be processed, as opposed to 257 unique
values for the RDFT case. This potentially represents a saving of roughly
50% in computational complexity.
Second, the reconstruction procedure suggests that the real (or imaginary) part of the DFT alone is sufficient
for perfect signal reconstruction, as long
as it is densely sampled in the frequency
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March 2018

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axis; therefore, it captures all the information of the discrete-time signal x (n).
A similar procedure can be derived to
perfectly reconstruct a signal from the
imaginary part of the DFT using the
same arguments. It can be inferred that
after the DFT is performed, the signal information is replicated and then
embedded among its RI parts. This
raises a particularly interesting question of whether processing just the
RDFT would be advantageous when
compared with processing both RI
parts, especially given the increasing
interest in incorporating phase-related
information in speech processing (such
as [8]).
Another real-valued transform is the
discrete cosine transform (DCT), which
can be interpreted as the DFT of a symmetrically extended signal. We have
compared the RDFT with the DCT in
modulation domain speech enhancement experiments and will report the
results in an upcoming paper.

Conclusions
In this tutorial, we have described a
procedure for the reconstruction of a
discrete-time, complex-valued signal
from the RDFT. The RDFT procedure
(continued on page 174)
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
