zeropadding

A remark on zero-padding for increased frequency
resolution
Fredrik Lindsten
November 4, 2010
1 Introduction
A common tool in frequency analysis of sampled signals is to use zero-padding
to increase the frequency resolution of the discrete Fourier transform (DFT).
By appending artificial zeros to the signal, we obtain a denser frequency grid
when applying the DFT. At first this might seem counterintuitive and hard to
understand. It is the purpose of this document to give a time-domain explana-
tion of zero-padding, which hopefully can help to increase the understanding of
why we obtain the results that we do.
2 Truncated DTFT
The discrete-time Fourier transform (DTFT) of a discrete-time signal x is given
by
XT (eiωT
) = T
∞
k=−∞
x[k]e−iωkT
. (1)
Now, assume that we have observed/measured N values of the signal x, i.e., we
know
x[0], x[1], . . . , x[N − 1], (2)
but apart from this the signal is unknown. This, very realistic scenario, leads to
an immediate difficulty when computing the DTFT, since the summation in (1)
ranges over unknown values of x. To be able to cope with this problem we need
to make some assumption about the behavior of the signal outside the known
range.
One possibility is to assume that the unknown values are zero, as illustrated
in Figure 1. If we make use of this assumption when computing the DTFT we
obtain the truncated DTFT,
X
(N)
T (eiωT
) = T
N−1
k=0
x[k]e−iωkT
. (3)
It is important to remember that, if the “true” signal x is non-zero outside the
sampled range, then the truncated DTFT is an approximation of the “true”
DTFT.
1

−10 −5 0 5 10 15 20
−1.5
−1
−0.5
0
0.5
1
1.5
2
Time (sample number)
? ?
−10 −5 0 5 10 15 20
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 1: (Left) Sampled signal, the values outside the sampled range are un-
known. (Right) One possible approximation, we assume that the signal is zero
outside the sampled range.
3 DFT
The discrete Fourier transform (DFT) is often presented as a discrete approx-
imation of the truncated DTFT. In other words, since the (truncated) DTFT
is frequency continuous (it is a function of the continuous variable ω), it is
problematic to represent and to work with using computers. Hence, to obtain
something that is manageable, we “sample” the truncated DTFT at a discrete
set of frequencies. This, frequency discrete, representation is the DFT of the
signal.
However, it is possible make another interpretation of the DFT. To do so we
return to the original problem, namely that we need to make some assumption
about the behavior of the signal x outside the known range. Now, instead of
assuming that it is zero, as we did for the truncated DTFT, we assume that
the signal is periodic with period N. This assumption is illustrated in Figure 2.
Hence, we claim that periodicity of the signal will turn the DTFT into the DFT.
Unfortunately, to show this is not as straightforward as plugging the periodic
signal into the definition of the DTFT (1) and do the computations. The reason
is that a periodic signal is not of bounded energy (not in 1), meaning that its
DTFT does not exist! We shall throughout this section still try to motivate the
proposition that periodicity will “turn the DTFT into the DFT”.
To start with, recall the definitions of the DFT and the inverse DFT, re-
spectively,
X[n] =
N−1
k=0
x[k]e− 2πi
N nk
, (4a)
x[k] =
1
N
N−1
n=0
X[n]e
2πi
N nk
, (4b)
n, k = 0, . . . , N − 1. (4c)
As indicated by (4c), the definition is only valid for n and k ranging from 0 to
N − 1. However, if we relax this condition, implicit periodicity of the signal
2

−10 −5 0 5 10 15 20
−1.5
−1
−0.5
0
0.5
1
1.5
2
? ?
−10 −5 0 5 10 15 20
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 2: (Left) Sampled signal, the values outside the sampled range are un-
known. (Right) One possible approximation, we assume that the signal is peri-
odic with period N.
can be seen already in the DFT deﬁnition. More precisely, let ¯k be any natural
number and decompose it as ¯k = Np + k for k ∈ [0, N − 1] and p a natural
number. Consider,
x[¯k] =
1
N
N−1
n=0
X[n]e
2πi
N n(Np+k)
=
1
N
N−1
n=0
X[n]e
2πi
N nk
e
2πi
N nNp
=1
= x[k]. (5)
Hence, writing the signal as in (4b) for arbitrary natural numbers k implicitly
implies that it is periodic. Analogously, the DFT in (4a) is also periodic with
period N.
Even though the implicit periodicity of the signal is indicated by the DFT
deﬁnition alone, we still want to show the relationship between the DTFT and
the DFT for periodic signals. As already pointed out, this does not allow for a
rigorous mathematical treatment, since the DTFT does not exist. However, to
circumvent this we shall make use of the standard convention that the DTFT
of a periodic signal can be written in terms of Dirac δ-“functions”. For an
introduction to this concept, see e.g., [1] Appendix 2.A. More precisely, we shall
make use of the DTFT of the constant function 1 expressed as (see [1], page
79),
DTFT{1} = T
∞
k=−∞
e−iωkT
= 2π
∞
n=−∞
δ ω −
2πn
T
. (6)
Now, consider the DTFT of the signal x, assumed to be periodic,
XT (eiωT
) = T
∞
k=−∞
x[k]e−iωkT
= T
∞
l=−∞
lN+N−1
k=lN
x[k]e−iωkT
= T
∞
l=−∞
N−1
j=0
x[lN + j]e−iω(lN+j)T
= T
N−1
j=0
x[j]e−iωjT
∞
l=−∞
e−iωlNT
.
(7)
3

For the second equality, we simply split the summation interval into blocks of
length N. We then make a change of summation index (j = k − lN). Finally,
we change the order of summation and make use of the assumed periodicity of
x. Using (6) (with ω replaced by Nω) we get
XT (eiωT
) = 2π
N−1
j=0
x[j]e−iωjT
∞
n=−∞
δ Nω −
2πn
T
=
2π
N
∞
n=−∞
δ ω −
2πn
NT
N−1
j=0
x[j]e−iωjT
. (8)
Here, we have made use of the scaling property δ(aω) = δ(ω)/|a|. The presence
of the δ-function turns the frequency continuous DTFT into a frequency discrete
representation, by “cutting out” a set of discrete frequency values. Hence, we
can replace ω by 2πn/NT, yielding
XT (eiωT
) =
2π
N
∞
n=−∞
δ ω −
2πn
NT
N−1
j=0
x[j]e− 2πi
NT njT
=
2π
N
∞
n=−∞
δ ω −
2πn
NT
X[n], (9)
where we have made use of the DFT definition (4a) (periodically extended to
all natural numbers n).
We can thus view the DTFT of a periodic signal as a frequency pulse train,
with δ-functions spread on a discrete frequency grid ω = 2πn/NT, n ∈ N.
Furthermore, each pulse is weighted with the DFT value at the corresponding
frequency.
As a final remark of this section, there is an analogy to the relationship
between the DTFT and the DFT for continuous-time signals. The DTFT, gen-
erally used for non-periodic signals, corresponds to the Fourier transform (FT)
in continuous time. The DFT on the other hand, corresponds to the Fourier
series (FS) in continuous time, and both apply to periodic signals. Just as in
continuous time the periodicity of the signal will turn the frequency transform
discrete, i.e., both the DFT and the FS are frequency discrete. Also, just as we
have expressed the DTFT of a periodic signal as a Dirac sum weighted by the
DFT, the FT of a continuous-time periodic signal is often expressed as a Dirac
sum weighted by the FS coefficients.
4 Zero-padding - a time domain explanation
In the previous section we concluded that, when computing the DFT of a signal
x[n], n = 0, . . . , N − 1, we implicitly assume that the signal is periodic with
period N. This insight can help us to understand how we can increase the
frequency resolution by using zero-padding. Assume that we create a new signal
by zero-padding x according to,
y[n] =
x[n], n = 0, . . . , N − 1,
0, n = N, . . . , M − 1.
(10)
4

−20 −10 0 10 20 30 40
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 3: Zero-padded signal y, with the implicit assumption that it is periodic.
Here, M = 2N = 24.
Hence, we simply augment x with M − N zeros (of course M > N). From the
deﬁnition of the truncated DTFT (3), it is clear that
Y
(M)
T (eiωT
) ≡ X
(N)
T (eiωT
). (11)
However, it is not true that x and y have the same DTFT (only their truncated
DTFTs coincide).
Now, if we compute the DFT of y, we make the implicit assumption that it is
periodic with period M, as illustrated in Figure 3. As more and more zeros are
appended to the signal, the periodicity assumption will more and more resemble
the assumption that the signal is zero outside the sampled range. Hence, by
padding the signal with zeros we “move” from the DFT assumption (periodicity)
to the truncated DTFT assumption (that the signal is zero outside the known
range). Consequently, the DFT of the signal will “move” toward the truncated
DTFT, as illustrated in Figure 4.
Hence, zero-padding will indeed increase the frequency resolution. However,
we do not gain any more information, we simply move from one assumption
to another. If the (non-truncated) DTFT of x is thought of as the truth, i.e.,
what we really seek, then zero-padding will not necessarily be of any help. It all
depends on which one of the two assumptions (periodicity or truncation) that
most closely resembles the “true” signal.
With that being said, we should not understate the use of zero-padding,
as it quite often can reveal (but not create) useful information about a signal.
However, as a user it is important to understand what is implicitly done when
we apply the tools of our toolbox. If we have this understanding, we are able
to apply the tools only when it is appropriate to do so, and also to draw sound
conclusions about the results that we obtain.
5

0 2 4 6
0
1
2
3
4
5
6
7
0 2 4 6
0
1
2
3
4
5
6
0 2 4 6
0
1
2
3
4
5
6
7
Frequency (rad/s)
0 2 4 6
0
1
2
3
4
5
6
7
Frequency (rad/s)
Figure 4: Zero-padding “moves” the DFT toward the truncated DTFT (the
absolute values of the transforms are displayed in all plots). (Top left) Truncated
DTFT of signal x. (Top right) DFT of signal x consisting of N = 12 samples.
(Bottom left) DFT of the same signal, zero-padded to double length. (Bottom
right) DFT of the same signal, zero-padded to three times the length.
6

References
[1] Fredrik Gustafsson, Lennart Ljung, and Mille Millnert. Signal Processing.
Studentlitteratur, Lund, Sweden, 2010.
7

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