L. Zhang et al.
APPENDIX: PROPOSITIONS AND PROOFS
Recall that the two-dimensional mapping of a N -dimensional data point is defined by real and imaginary components
of:
N
−1
N
−1
X
X
n
F1 (x[n]) =
x[n]WN
=
x[n]e−i2πn/N .
(1)
n=0
−i2π/N
where WN = e
n=0
is called twiddle factor, 2π/N is the base frequency.
Lemma 1 (Conjugate) For any two complex numbers z and w, (1) z + w = z + w, (2) z w = z w, (3) z = z.
Lemma 2 (Square Expanding)
ÃN −1
X
!2
an
=
N
−1
X
n=0
a2n
+2
n=0
Lemma 3 (Cancellation) Let j ∈ N, then
N
−1
X
e−i2πjn/N =
n=0
−k−1
N
−1 N X
X
N
−1
X
at at+k .
t=0
k=1
cos(2πjn/N ) =
n=0
N
−1
X
sin(2πjn/N ) = 0.
n=0
Proof:
N
−1
X
e−i2πjn/N =
n=0
then apply e
iθ
1−1
e−i2πjN/N − 1
= −i2π/N
= 0.
−i2π/N
e
−1
e
−1
= cos θ + i sin θ, we get
N
−1
X
e
−i2πjn/N
n=0
=
N
−1
X
cos(2πjn/N ) − i
n=0
N
−1
X
sin(2πjn/N ) = 0 + 0i.
n=0
Lemma 4 (Homomorphism) FFHP is homomorphic. F1 (a x[n] + b y[n]) = a F1 (x[n]) + b F1 (y[n]).
From the formula in Eq. (1), we get
F1 (a x[n] + b y[n]) =
N
−1
X
(a x[n] + b y[n])e−i2πn/N = a
n=0
=
N
−1
X
x[n]e−i2πn/N + b
n=0
N
−1
X
y[n]e−i2πn/N
n=0
a F1 (x[n]) + b F1 (y[n]).
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Proposition 1 (Cancellation) An N -dimensional point with equal dimension values will be mapped onto the origin. If x[n] = [a, . . . , a], then F1 (x[n]) = 0.
Proof : From the formula in Eq. (1), we get
F1 (x[n]) =
N
−1
X
n=0
By Lemma 3, let j = 1. F1 (x[n]) = 0.
a e−i2πn/N = a
N
−1
X
e−i2πn/N .
n=0
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L. Zhang et al.
Proposition 2 (Addition by a Constant) Two N -dimensional points with addition of a constant to each dimension
value will be mapped onto the same point. If y[n] = x[n] + a, then F1 (y[n]) = F1 (x[n]).
Proof : From the formula in Eq. (1), we get
F1 (y[n])
=
N
−1
X
(x[n] + a)e−i2πn/N =
N
−1
X
n=0
=
(x[n] e−i2πn/N + a e−i2πn/N )
n=0
N
−1
X
x[n] e
−i2πn/N
+a
n=0
N
−1
X
e
−i2πn/N
= F1 (x[n]) + a
n=0
N
−1
X
e−i2πn/N
n=0
By Proposition 1, the second summation is 0.
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Proposition 3 (Scaling by a Constant) Two N -dimensional points with scaling of a constant to each dimension
value will be mapped onto two points on a line through the origin. If y[n] = a x[n], then F1 (y[n]) = a F1 (x[n]).
Proof : From the formula in Eq. (1), we get
F1 (y[n]) =
N
−1
X
a x[n]e−i2πn/N = a
n=0
N
−1
X
x[n]e−i2πn/N = a F1 (x[n])
n=0
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Proposition 4 (Time Shifting) Two N -dimensional points with constant time shifting will be mapped onto a circle
concentric to the unit circle. The angle between two images is d2π/N . If y[n] = x[n − d], then F1 (y[n]) =
d
F1 (x[n])WN
.
Proof : Assume 0 ≤ n < N , let l = n − d, then n = l + d. When n = 0, l = −d and when n = N − 1,
l = N − 1 − d. From the formula in Eq. (1), we get
F1 (y[n])
=
F1 (x[n − d]) =
NX
−1−d
x[l]e−i2π(l+d)/N
l=−d
=
NX
−1−d
d
x[l]e−i2πl/N e−i2πd/N = WN
l=−d
NX
−1−d
x[l]e−i2πl/N
l=−d
However, ei2πn/N = ei2π(n+N )/N and x[n] = x[n + N ],
NX
−1−d
x[l]e−i2πl/N =
l=−d
−1
X
x[l + N ] e−i2π(l+N )/N +
l=−d
NX
−1−d
x[l] e−i2πl/N
l=0
Let t = l + N for the first summation and t = l for the second summation, we get
NX
−1−d
l=−d
x[l]e−i2πl/N =
N
−1
X
x[t]e−i2πt/N +
t=N −d
d
Therefore, F1 (y[n]) = F1 (x[n])WN
.
NX
−1−d
t=0
x[t]e−i2πt/N =
N
−1
X
x[t]e−i2πt/N = F1 (x[n])
t=0
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L. Zhang et al.
Proposition 5 (Line) Under FFHP, an N -dimensional line will be mapped onto a two dimensional.
Proof : A N -dimensional line l through point P and parallel to a N -vector ∆ can be expressed as P + t∆ where
−∞ ≤ t ≤ ∞. Let Q and R are two different points on l, then Q = P + α∆ and R = P + β∆, for some
α, β ∈ R. Let the corresponding signals for P , Q, R, and ∆ be p[n], q[n], r[n], and δ[n]. By Lemma 4, F1 (q[n]) =
F1 (p[n] + αδ[n])) = F1 (p[n]) + αF1 (δ[n]) and F1 (r[n]) = F1 (p[n]) + βF1 (δ[n]). Compare the definition of a
line above, F1 (q[n]) and F1 (r[n]) are two points on a two dimensional line through F1 (p[n]) and parallel to the
vector F1 (δ[n]).
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Definition 1 (Mean, Autocovariance, Variance, Autocorrelation Coefficient) The mean of a signal x[n] is dePN −1
fined as x
b =
n=0 x[n]/N . The k-th sample autocovariance coefficient of a signal x[n] is defined as gk =
PN −1−k
(x[n] − x
b)(x[n + k] − x
b)/N . g0 is called the variance of x[n]. The k-th sample autocorrelation coeffin=0
cient is defined as rk = gk /g0 .
Proposition 6 (Fundamental Distance) Let w[n] = x[n] − y[n], be the difference between x[n] and y[n]. The
distance between F1 (x[n]) and F1 (y[n]) is
Ã
2
kF1 (w[n])k = g0 N
1+2
N
−1
X
!
rk cos(2πk/N ) .
k=1
Proof : By Lemma 4, the distance between F1 (y[n]) and F1 (x[n]) is kF1 (w[n])k. From Eq. (1), we get
kF1 (w[n])k = k
N
−1
X
w[n]e−i2πn/N k = k
n=0
N
−1
X
Let ω = 2π/N , by Lemma 3, we have
N
−1
X
cos(nω) =
n=0
w[n],
kF1 (w[n])k2
=
ÃN −1
X
ÃN −1
X
N
−1
X
sin(nω) = 0. Now add a term w,
b the mean of
n=0
!2
w[n] cos(nω)
n=0
=
w[n] cos(2πn/N ) − iw[n] sin(2πn/N )k
n=0
+
ÃN −1
X
!2
w[n] sin(nω)
n=0
!2
(w[n] − w)
b cos(nω)
+
ÃN −1
X
n=0
!2
(w[n] − w)
b sin(nω)
n=0
Expending each squaring terms by Lemma 2, we get
N
−1
X
(w[n] − w)
b 2 (cos2 (nω) + sin2 (nω)) + 2
n=0
N
−1 N X
−1−k
X
k=1
[(w[t] − w)(w[t
b
+ k] − w)∆]
b
t=0
where ∆ = cos(tω) cos((t + k)ω) + sin(tω) sin((t + k)ω). By trigonometry identity cos θ cos φ + sin θ sin φ =
cos(φ − θ), ∆ = cos(kω). Now we have
kF1 (w[n])k2
=
N
−1
X
(w[n] − w)
b 2+2
n=0
=
N (g0 + 2
N
−1 N X
−1−k
X
k=1
N
−1
X
k=1
[(w[t] − w)(w[t
b
+ k] − w)
b cos(kω)]
t=0
gk cos(kω)) = g0 N
Ã
1+2
N
−1
X
!
rk cos(2πk/N )
k=1
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L. Zhang et al.
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Definition 2 (Twiddle Power Index) For an N -point signal, the k-th harmonic twiddle power index (HTPI in
short) is a sequence of N time indices chosen from 0, . . . , N − 1. It corresponds to the order that a particular
N −1
0
time point being mapped on to the consecutive powers of twiddle factor WN
,. . . , WN
, by the k-th harmonic.
Example 1 For a 5-point signal, the first harmonic twiddle power index is [0, 1, 2, 3, 4]. The second HTPI is
PN −1
2k
[0, 3, 1, 4, 2]. The third HTPI is [0, 2, 4, 1, 3]. Take a closer look at the second HTPI. Since F2 (x[n]) = n=0 x[n]WN
,
0
2
4
1
3
0
1
2
3
we have F2 (x[n]) = x[0]W5 +x[1]W5 +x[2]W5 +x[3]W5 +x[4]W5 = x[0]W5 +x[3]W5 +x[1]W5 +x[4]W5 +
x[2]W54 .
Proposition 7 (Harmonic Equivalence) A k-th harmonic of a signal (k > 1) is equivalent to the first harmonic of
the the origin signal being reordered by the k-th harmonic twiddle power index.
Proof: By definition of harmonic and twiddle power index.
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