IEEE Signal Processing - March 2018 - 163

x p (n )

x (n )

n

n
(a)

(b)
1
[x (n ) + xp∗ (-n )]
2 p

x p∗ (-n )

n

n
(c)

(d)

Figure 1. Plots show the reconstruction of a discrete-time signal (without zero-padding) from only the RDFT: (a) the original real-valued signal x (n) with
no delay or zero-padding (N = 4); (b) a periodic version of x (n ) (with a period of N ) as a result of the cyclic properties of the DFT; (c) a complex conjugate and time-reversed version of x p (n); and (d) an output signal from taking the inverse DFT of the real part, showing the effect of time-domain aliasing.
The shaded region shows the reconstructed signal.

yu (n) = 1
M

M- 1

/

2 Xu R (k) e j2rnk/M .

k= 0

for n = 0, 1, f, M - 1

(4)

The resulting signal y (n) from the
inverse DFT contains the original signal
x (n) embedded within it [see (5) in the
box at the bottom of the page], where
x) (n) is the complex conjugate of x (n).
Therefore, the original discrete-time
signal x (n) can be perfectly recovered
from this procedure.

Theoretical basis for the RDFT
The theoretical basis for the reconstruction procedure that was described in
the previous section comes from a DFT
property from [4, eq. (8.110)] [see (6) in
the box at the bottom of the page], where
x p (n) is a periodic signal (with a period
of N ), where each period is identical
to x (n), and Re {X (k)} is the RDFT
of x (n). When looking at this prop-

erty, recovering x (n) from the RDFT
alone does not appear possible without
the imaginary part . A similar situation
occurs in the corresponding property
that relates the imaginary part of the DFT
as follows [4]:
1 [x (n) - x * (- n)]
p
p
2

DFT

jIm{X (k)}.

This is because of the presence of the
complex conjugate term x *p (- n), which
completely overlaps with x p (n), and
therefore results in time-domain aliasing.
This is shown in Figure 1 for the case of a
real-valued discrete-time signal of length
N = 4. Figure 1(d) shows the final signal as a result of taking the inverse of the
RDFT of x (n) [shown in Figure 1(a)]. We
can see the effects of aliasing distortion in
Figure 1(d). Therefore, the original signal
x (n) cannot be perfectly recovered from
the RDFT in this case.

for n = 1, 2, f, N
x (n - 1),
yu (n) = * x) (M - 1- n), for n = M - N, f, M - 1,
elsewhere
0,

(5)

1 [x (n) + x * (- n)]
p
p
2

(6)

DFT

R e {X (k)}

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March 2018

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To circumvent the problem of timedomain aliasing, the signal x (n) is
delayed by one sample and a zero is padded at the beginning to avoid the overlap
at n = 0. Further padding of N zeros is
performed at the end of the signal. This
effectively extends the period of x p (n)
from N to 2N + 1. Therefore, by choosing the value of M $ 2N + 1, we can
prevent the time-domain aliasing. This is
demonstrated in Figure 2 for a real-valued
discrete-time signal of length N = 4. As
we can see in Figure 2(d), the original signal x (n) can be perfectly recovered from
the RDFT if the appropriate delay and
zero-padding are applied.

Discussion and applications
One potential application of the RDFT
method of signal reconstruction is in
the area of speech enhancement, where
noise-corrupted speech is processed to
alleviate the degrading effects of the
noise and therefore improve the quality and intelligibility of the speech.
Recent studies in this area [5], [6] have
shown that, over time, the estimation of
the RI DFT coefficients are as effective as modulation-magnitude domain
processing, where the DFT magnitude
coefficients are temporally processed
and then combined with the noisy DFT
163
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
