On Hilbert Transform of Signals on Graphs
Arun Venkitaraman, Saikat Chatterjee, Peter Händel
Department of Signal Processing
School of Electrical Engineering and ACCESS Linnaeus Center
KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
arunv@kth.se, sach@kth.se, ph@kth.se
Abstract—We propose definitions for Hilbert transform and
analytic signal construction for real signals on graphs using the
graph Fourier transform (GFT). The definitions are based on the
conjugate-symmetry-like property exhibited by the GFT basis
of a graph with real-valued adjacency matrix. We show that a
real graph signal (GS) can be represented using smaller number
of GFT coefficients than the signal length, leading to notions of
graph analytic signal (GAS) and graph Hilbert transform (GHT),
which include their conventional counterparts as special cases.
We prove that the GHT and GAS operations are linear and shiftinvariant on graphs. We also propose definitions for amplitude,
phase, and frequency modulations for the GS, and discuss phaseunwrapping for graph signals. We illustrate the concepts using
synthesized and real-world signal examples.
Index Terms—Graph signal, analytic signal, Hilbert transform,
demodulation, graph phase-unwrapping.
I. I NTRODUCTION
Data processing on large datasets has generated significant
interest recently thanks to the data deluge experienced by an
increasing number of scientific disciplines. Advances in smart
device technology and networked applications have resulted in
an increased dimensionality and diversity of generated data,
posing new challenges in the analysis of large dimensional
data, particularly, given their networked or connected nature.
It then becomes of interest to look into how existing signal
processing methodologies may be extended to facilitate our
understanding of such data, while making use of the connectivity information. Such efforts have led to the emergence of
the notion of signal processing on graphs [1].
Signals arising in a large variety of applications, such
as social and economic networks, epidemiology, biological
networks, transportation networks, internet blog data, etc.,
may be modeled as signals on graphs characterized by a
connectivity or adjacency matrix that captures dependencies in
the data [1], [2]. A number of concepts from standard discretetime signal processing (DSP) have been extended to the graph
signal paradigm recently. Wavelet transforms and multiresolution representations have been proposed for the modeling
and analysis of distributed data and sensor networks [3]–[6].
Filterbank concepts such as perfect-reconstruction and critical
sampling have also been considered for signals on graphs
[7]. Recently, Thanou et al. proposed a parametric dictionary
learning approach for graph signals [8]. A majority of the
works mentioned so far are based on concepts from spectral
The authors would like to acknowledge the support received from the
Swedish Research Council.
graph theory [9], which uses the graph-Laplacian as the central
unit, and are thereby restricted to the analysis of undirected
graphs [1]. An alternative approach to graph signal processing,
as proposed by Sandryhaila and Moura [2], is to define a shift
or translation of the graph signal using the adjacency matrix,
and arrive at notions of graph linear filtering and graph Fourier
transform (GFT). The approach uses concepts from algebraic
signal processing and differs from the graph-Laplacian based
approach [2], [10]. In particular, the approach is applicable
to signals on arbitrary directed or undirected graphs, with
possibly complex-valued adjacency matrix [11].
In this paper, we propose definitions for the Hilbert transform and analytic signal for real signals on graphs, based
on the graph signal framework proposed in [2]. We shall
hereafter refer to real signals on a graph as real graph signals.
We show that a real graph signal (GS) on a graph with a
real adjacency matrix may be represented using a smaller
number of GFT coefficients than the signal length, akin to
the ‘one-sided’ spectrum for conventional one-dimensional (1D) signals. The definitions use the conjugate-symmetry-like
property of the GFT basis. We show that the proposed graph
Hilbert transform (GHT) and graph analytic signal (GAS)
operations are linear and shift-invariant on graphs. Using the
graph analytic signal, we propose definitions for the amplitude,
phase, and frequency modulations of a GS. We discuss the
phase-unwrapping operation for graph signals and develop two
plausible algorithms. We show the application of the proposed
concepts to synthesized signals and real-world speech signals.
II. P RELIMINARIES
A. The graph signal
Let x ∈ RN be a real signal on the graph G = (V, A),
where V and A denote the vertex set and the adjacency matrix,
respectively. Then, the graph Fourier transform Fg of x is
defined as [2], [11]:
x̂ , [x̂(1), x̂(2), . . . , x̂(i), . . . , x̂(N )]> = Fg {x} = V−1 x,
(1)
where V denotes the eigenvector matrix such that
A = VJV−1 , and J the diagonal eigenvalue matrix
J = diag(λ1 , λ2 , · · · , λN ).
B. The standard analytic signal
Let x̂(ω) denote the discrete Fourier transform (DFT) of
the real 1-D signal x evaluated at frequency ω. Then, the
discrete analytic signal of x, denoted by xa,c , has the following
frequency-domain definition [13]–[15]:

2π
2π


2x̂(ω), ω ∈ N , · · · , π − N
(2)
x̂a,c (ω) = x̂(ω), ωn∈ {0, π}
o


0, ω ∈ π + 2π , · · · , 2π(N −1) ,
N
N
where the subscript c denotes the conventional (standard)
definition. Taking the inverse DFT on
√ both sides of (2), we
get that xa,c = x + jxh,c , where j = −1 and xh,c is known
as the discrete Hilbert transform (DHT) of x [13]. We note
that an N -sample 1-D signal
a GS x with
 can be seen as 

the adjacency matrix C = 
0
0
.
.
.
1
1
0
.
.
.
0
0
1
.
.
.
0
···
···
.
.
.
···
0
0
0
III. G RAPH A NALYTIC S IGNAL
We assume A to be real, which means all its eigenvalues
and the corresponding eigenvectors are either real or appear in
conjugate-pairs. We assume further that A is diagonalizable.
We sort the eigenvalues in the ascending order of their phase
angle from 0 to 2π to form the diagonal matrix J, and
correspondingly sort the eigenvectors such that A = VJV−1 .
If multiple eigenvalues with same phase angle occur, we order
them in the descending order of their magnitude. Let K1 and
K2 denote the number of real-valued positive and negative
eigenvalues of A, respectively, and K = K1 + K2 . Let us
define the sets:
Γ2
Γ3
Γ4
Definition (Graph analytic signal (GAS)). We define the graph
analytic 
signal of x as xa = Fg−1 {x̂a } = Vx̂a , where

2x̂(i), i ∈ Γ2
x̂a (i) = x̂(i), i ∈ Γ1 ∪ Γ3 .


0,
i ∈ Γ4
In the case when all the eigenvalues of A are complex (K =
0), number of non-zero coefficients in x̂a is exactly one half
of the total, resulting in a one-sided spectrum.
A. Graph Hilbert transform

, and in this
case, the GFT coincides with the DFT [12].
Γ1
conventional analytic signal construction in (2), we define the
graph analytic signal (GAS) as follows:
{1, · · · , K1 } (positive real eigenvalues)
N −K
=
K1 + 1, · · · , K1 +
2
(eigenvalues with phase angle in (0, π))
N −K
N +K
=
K1 +
+ 1, · · · ,
2
2
(negative real eigenvalues)
N +K
+ 1, · · · , N
=
2
(eigenvalues with phase angle in (π, 2π))
=
Then, our choice of ordering of the eigenvalues of A results
in the following structure on the graph Fourier coefficients:
x̂(i) = x̂∗ (N − i + K1 + 1), i ∈ Γ2 .
(3)
Since A is real, N and K are always of the same parity (odd
or even). In the case of a 1-D signal, that is, when A = C,
(3) reduces to the familiar conjugate-symmetry property of
the DFT [13]. This indicates that, similar to the 1-D case, a
real GS can be represented using θ GFT coefficients, where
θ = |Γ1 | + |Γ2 | + |Γ3 | = (N + K)/2 , and |Γ| denotes the
cardinality of the set Γ. For K << N , θ ≈ N/2. We note that
equation (3) holds if and only if x is real, which means that,
given the same graph, a graph signal which does not satisfy
(3) is necessarily complex-valued. Motivated by (3) and the
As a consequence of the one-sidedness of the GFT spectrum, we have that xa is complex and hence, is expressible as
xa = x + j xh . We define xh as the Graph Hilbert transform
(GHT) of x. Then, from the definition of the GAS, we have
that


+x̂(i), i ∈ Γ2
(4)
jx̂h (i) = 0,
i ∈ Γ1 ∪ Γ3


−x̂(i), i ∈ Γ4 .
Equation (4) generalizes the frequency-domain definition of
the DHT [13]. We next show that the GHT xh of real x is
real. Since Fg−1 {x̂h } = Vx̂h = xh , we have
P
P
jxh =
jx̂h (i)vi
i∈Γ2 jx̂h (i)vi +
P
P i∈Γ4
=
x̂(i)vi − i∈Γ4 x̂(i)vi
Pi∈Γ2
∗
∗
=
i∈Γ2 (x̂(i)vi − x̂ (i)vi )
P
(5)
= 2 j=
i∈Γ2 x̂(i)vi ,
where vi denotes the i’th column of V, and =(a) denotes the
imaginary part of a. The third equality in (5) follows from
the observation that eigenvectors indexed by Γ2 and Γ4 form
complex conjugates. Thus, jx̂h is purely imaginary, or xh is
real, which in turn means that x = <(xa ), where <(a) denotes
the real part of a.
Equation (4) can be expressed as x̂h = Jh x̂, where Jh is
the diagonal matrix whose i’th diagonal element is given by


−j, i ∈ Γ2
Jh (i) = 0,
(6)
i ∈ Γ1 ∪ Γ3


+j, i ∈ Γ4 .
Proposition 1. The GHT is a linear graph shift-invariant
operation, that is, for a GS x and a linear graph shift-invariant
filter M, we have that (Mx)h = M (xh ).
Proof. A linear
shift-invariant graph filter on G is of the
PL
i
= m(A) for some L ≤ N .
form M =
i=0 mi A
−1
Since A = VJV , we have that M = Vm(J)V−1 . Let
y denote the output of the filter M for the input x. Then,
by the convolution property of the GFT [12], we have that
ŷ = m(J)x̂. Since x̂h = Jh x̂, we get that
ŷh = Jh ŷ = Jh m(J) x̂ = m(J) Jh x̂ = m(J) x̂h ,
(7)
where we have used the commutativity of the diagonal matrices m(J) and Jh . Taking the inverse GFT on both sides
of (7), we get that (Mx)h = M (xh ), which completes the
proof.
Algorithm 1 Graph Phase Unwrapping 1
1: Set φu
x,V (1) = φx,V (1), and loc(1) = 1, Ω = {1, · · · , N }.
2: For 2 ≤ i ≤ N , set Ωi = {loc(1), · · · , loc(i − 1)}c , and
find:
loc(i) = argmax|A(loc(i − 1), j)|.
j∈Ωi
Since GHT is linear and graph shift-invariant and x̂h =
Jh x̂, using the convolution
property
of GFT, we have that
P
L
i
xh = h(A)x =
h
A
x,
such
that h(J) = Jh . In
i=0 i
other words, the GHT can be implemented as linear shiftinvariant graph filter whose coefficients hi are obtained by
solving the following system of linear equations:
h0 + h1 λi + · · · + hL λL
i = 0,
h0 + h1 λi + · · · +
h0 + h1 λi + · · · +
hL λL
i
hL λL
i
i ∈ Γ1 ∪ Γ3
= −j,
i ∈ Γ2
= +j,
i ∈ Γ4 .
(8)
The solution of (8) obtained by setting A = C and L = N is
the impulse response of the DHT.
IV. T HE GAS AND M ODULATION A NALYSIS
The analytic signal is used extensively in the demodulation of amplitude-modulated frequency-modulated (AM-FM)
signals [16]–[19]. The AM-FM model decomposes the signal
into two components: one varying smoothly, capturing the
average information or the envelope of the signal, referred
to as the AM, and the second, varying more rapidly, capturing
the finer variations in the signal, and referred to as the phase
or frequency modulation (PM or FM). Most demodulation
techniques involve the construction of the AS, implicitly or
explicitly. We next define the AM and PM for graph signals
by generalizing the standard 1-D definitions [15]–[17]:
Definition (Amplitude and phase modulation). The amplitude
and phase modulations of a GS x, denoted by Ax,V and φx,V ,
are defined as the magnitude and phase angle of the graph
AS, respectively:
Ax,V (i)
= |xa (i)| ,
φx,V (i)
=
∀ i ∈ {1, 2, · · · , N }
arg(xa (i)),
(9)
where xa (i) denotes the i’th component of the vector xa and
arg(·) denotes the 4-quadrant arctangent function which takes
values in the range (−π, π] by considering the quadrant in
which xa lies.
In the case when A = C, (9) reduces to conventional AM
and PM definitions [16], [17]. We next discuss the issue of
computing the unwrapped phase and the phase-derivative.
A. Phase unwrapping and frequency modulation
The arg(·) function returns phase values wrapped in the
range (−π, π]. In practice, it is more convenient to work
with unwrapped phase functions [20]. In the case of 1-D
signals, the phase-unwrapping (PU) is performed by keeping
the causality in mind: unwrapping begins from the first sample,
successively compensating for the step discontinuities in the
phase in a cumulative manner [13], [21]. PU algorithms for
high-dimensional signals are also based on phase discontinuity
3:
4:
φ0x,V (i) = φx,V (loc(i)).
Perform standard 1-D PU on φux,V (i) = unwrap(φ0x,V )(i).
compensation, though the exact strategy may depend on the
signal model and the type of application [22], [23].
The issue of unwrapping φx,V becomes challenging due to
the nature of the signal connectivity involved. In contrast with
the single path or connecting link among the signal nodes
in the case of a 1-D signal, each node of a general graph
may be connected to multiple nodes and it is desirable that
the phase-unwrapping algorithm incorporates such connectivity information in a meaningful way. We next propose two
potential strategies for unwrapping the phase of the GAS, and
both include the standard 1-D PU as a special case. Let A(i, j)
denote the (i, j)’th entry of A.
Approach 1: Starting from node 1, we search for the node
connected to 1 with the maximum edge-weight magnitude.
Let us denote this node by 20 . We next proceed to find the
node 30 most strongly connected to 20 , excluding node 1,
and continue till all the nodes are numbered to obtain the
sequence {1, 20 , · · · , N 0 }, assuming that it is possible to
traverse all the nodes in the graph. We construct the new
phase sequence φ0x,V (i) = φx,V (i0 ), to which we apply
standard 1-D PU to obtain φux,V (i). Algorithm 1 shows the
steps involved in the process. In the case when multiple
nodes connected to the current node have equal edge-weights,
we break the tie arbitrarily. We also note that the approach
implicitly assumes a directionality in the graph due to its
selectivity to strongly-connected edges.
Approach 2: We define a new phase sequence φ0x,V as
i
>
follows: φ0x,V (i) = e>
1 A φx,V , where e1 = (1, 0, · · · , 0) .
The new phase sequence is constructed by stacking the first
entries of all the graph-shifted versions of φx,V . We then apply
standard 1-D PU on φ0x,V to obtain the unwrapped phase φux,V .
Algorithm 2 is a generalization of the 1-D PU idea where we
collect the phase at node 1 and its successive graph-shifted
values, akin to using successive time samples in 1-D, and
perform a 1-D PU operation. We observe that unlike Algorithm
1, the values of φ0x,V (i), i > 1 obtained from Algorithm 2
are weighted linear combinations of entries of φx,V . We note
that unlike Algorithm 1, Algorithm 2 involves powers of A,
and may not lead to an intuitive unwrapped phase in cases
where the eigenvalues of A are of small magnitudes, as both
the eigenvalues and the rank of Ai decrease with i. We next
define frequency modulation for graph signals:
Definition (Frequency modulation). The frequency modulation
u
(FM) of a GS x is defined as ωx,V = φux,V − |λ|−1
max Aφx,V ,
u
where φx,V denotes the unwrapped phase of the graph AS.
Algorithm 2 Graph Phase Unwrapping 2
1: Compute φx,V using (9).
i
A
2: For 1 ≤ i ≤ N ,
φ0x,V (i) = e>
φx,V .
1
|λ|max
0
3: Perform standard 1-D PU on φu
(i)
=
unwrap(φ
x,V
x,V )(i).
The definition is a generalization of the backward difference
operator used to define the FM for 1-D signals [13], noting
that multiplication by A is defined as a unit graph-shift. The
division by |λ|max compensates for norm scaling introduced
by A [12]. We note that the proposed AM and PMs (and
the associated FMs) obtained from both PU algorithms form
unique invertible representations of the GS only if I−|λ|−1
max A
is invertible. In addition, the unwrapping operation in Algorithm 2 is invertible if and only if all powers of A have full
rank. In contrast, the AM-FM representation obtained using
Algorithm 1 requires only the FM operation I − |λ|−1
max A to
be invertible, as φx,V can always be obtained by a reverse
permutation of φ0x,V .
(a)
(b)
(c)
Fig. 1. Non-uniformly sampled 1-D signal. Signal length N = 200.
V. E XPERIMENTS
We show the application of the proposed definitions to
synthesized signals and real-world speech signals. We first
consider a synthesized signal example. As noted earlier, a
GS with A = C represents a standard 1-D signal; the unit
edge-weights in C denote uniformly spaced samples. Using
a similar argument, a non-uniformly sampled 1-D signal
may be modeled as a GS
 with the adjacency matrix A =



0
0
.
.
.
wN
w1
0
.
.
.
0
0
w2
.
.
.
0
···
···
.
.
.
···
0
0

, where wi denotes the spacing
0
between the i’th and (i + 1)’th samples. We construct the
GAS for the real part of eigenvectors of A, where wi s are
drawn from uniform distribution over (0, 1). We compute the
graph AM |xa | and the AM obtained from the standard AS,
hereafter referred to as the ‘1-D AM’. From our experiments,
we have found that in comparison with the 1-D AM, the graph
AM is generally a better choice for the signal envelope as it
fits the signal more closely while preserving the onset and tail
decay characteristics. Figure 1 shows results for a particular
realization corresponding to the tenth eigenvector.
We next consider the GS x obtained from the diffusion
of a sparse signal x0 , that is, x = Ax0 , where A is the
Fig. 2. One-step graph diffusion for signal length N = 200. (a) kxk0 = 5,
(b) zoomed-in plot of (a), (c) kxk0 = 10. The red curve indicates the nodes
at which x0 is non-zero.
adjacency matrix obtained by orthonormalization of a matrix
with rows drawn from a standard normal distribution. Over
various realizations, we observe that when the number of nonzero entries in x0 , denoted by kx0 k0 , is small in comparison
with N , |xa | carries peaks at the locations of the non-zero
entries of x0 (diffusion source locations), whereas the 1-D
AM, computed by treating x as a 1-D signal, does not show
such a characteristic. This suggests that the source nodes may
be identified by peak-picking of |xa |. As kx0 k0 is increased,
|xa | continues to have peaks at the source locations, though
additional peaks are obtained elsewhere. Such peak-selectivity,
however, is not observed for x = AK x0 , for K > 1. We
show the plots for a particular realization in Figure 2.
We next consider application on speech signals taken from
[24]. We construct A by connecting every sample to its
succeeding P samples with edge-weights equal to P1 . Let
FM1 and FM2 denote the FMs obtained from PMs computed
using Algorithms 1 and 2, respectively. As P increases, FM1
becomes smoother and takes values closer to mean of FM
computed using the standard AS (1-D FM), whereas FM2 is
near-zero everywhere except for the initial few samples. This
is so because the values of entries and the rank of Ai both
decrease with i, resulting in a poor unwrapped PM, and hence,
phase remains open and it would be interesting to investigate
into possible alternatives than the ones presented here. We
hope to work along these directions in the future.
R EFERENCES
(a)
(b)
(c)
Fig. 3. Speech signal, female utterance of the word ’Head’, sampled at
16 kHz. (a) AM, and (b) FM for P = 2. and (c) FM for P = 8.
a poor FM. The AM does not exhibit significant variation over
P . In Figure 3, we show the graph AM and FM for different
values of P , for a particular speech segment.
VI. C ONCLUSION
We proposed definitions for the analytic signal and Hilbert
transform of real graph signals using the conjugate-symmetrylike property exhibited by the GFT basis. We showed that GHT
and GAS operations are linear and shift-invariant and that the
GHT can be represented as a linear shift-invariant graph filter.
Using the GAS, we defined amplitude, phase, and frequency
modulations for graph signals, and proposed two approaches
for generalization of phase-unwrapping operation to graph
signals. The proposed concepts were shown to reduce to their
standard 1-D counterparts as a special case. We considered
synthesized and real-world signal examples to illustrate the
proposed concepts.
The type of signals chosen for examples in this article only
illustrate the ideas developed and are by no means exhaustive.
As with conventional modulation analysis, the utility of the
proposed concepts will vary from application to application
and can only be revealed by detailed analysis on various
datasets. Also, the problem of computing an unwrapped graph
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