Introduction to Fourier transform and signal analysis

Continuous Fourier transform
Discrete Fourier transform
References
Introduction to Fourier transform and signal
analysis
Zong-han, Xie
icbm0926@gmail.com
January 7, 2015
Zong-han, Xieicbm0926@gmail.com Introduction to Fourier transform and signal analysis

References
License of this document
Introduction to Fourier transform and signal analysis by Zong-han,
Xie (icbm0926@gmail.com) is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International License.

References
Outline
1 Continuous Fourier transform
2 Discrete Fourier transform
3 References

References
Orthogonal condition
Any two vectors a, b satisfying following condition are
mutually orthogonal.
a∗
· b = 0 (1)
Any two functions a(x), b(x) satisfying the following
condition are mutually orthogonal.
a∗
(x) · b(x)dx = 0 (2)
* means complex conjugate.

References
Complete and orthogonal basis
cos nx and sin mx are mutually orthogonal in which n and m
are integers.
π
−π
cos nx · sin mxdx = 0
π
−π
cos nx · cos mxdx = πδnm
π
−π
sin nx · sin mxdx = πδnm (3)
δnm is Dirac-delta symbol. It means δnn = 1 and δnm = 0
when n = m.

References
Fourier series
Since cos nx and sin mx are mutually orthogonal, we can expand
an arbitrary periodic function f (x) by them. we shall have a series
expansion of f (x) which has 2π period.
f (x) = a0 +
∞
k=1
(ak cos kx + bk sin kx)
a0 =
1
2π
π
−π
f (x)dx
ak =
1
π
π
−π
f (x) cos kxdx
bk =
1
π
π
−π
f (x) sin kxdx (4)

References
Fourier series
If f (x) has L period instead of 2π, x is replaced with πx/L.
f (x) = a0 +
∞
k=1
ak cos
2kπx
L
+ bk sin
2kπx
L
a0 =
1
L
L
2
− L
2
f (x)dx
ak =
2
L
L
2
− L
2
f (x) cos
2kπx
L
dx, k = 1, 2, ...
bk =
2
L
L
2
− L
2
f (x) sin
2kπx
L
dx, k = 1, 2, ... (5)

References
Fourier series of step function
f (x) is a periodic function with 2π period and it’s deﬁned as
follows.
f (x) = 0, −π < x < 0
f (x) = h, 0 < x < π (6)
Fourier series expansion of f (x) is
f (x) =
h
2
+
2h
π
sin x
1
+
sin 3x
3
+
sin 5x
5
+ ... (7)
f (x) is piecewise continuous within the periodic region. Fourier
series of f (x) converges at speed of 1/n.

References
Fourier series of step function

References
Fourier series of triangular function
f (x) is a periodic function with 2π period and it’s deﬁned as
follows.
f (x) = −x, −π < x < 0
f (x) = x, 0 < x < π (8)
f (x) =
π
2
−
4
π
n=1,3,5...
cos nx
n2
(9)
f (x) is continuous and its derivative is piecewise continuous within
the periodic region. Fourier series of f (x) converges at speed of
1/n2.

References
Fourier series of triangular function

References
Fourier series of full wave rectiﬁer
f (t) is a periodic function with 2π period and it’s deﬁned as
follows.
f (t) = − sin ωt, −π < t < 0
f (t) = sin ωt, 0 < t < π (10)
f (t) =
2
π
−
4
π
n=2,4,6...
cos nωt
n2 − 1
(11)
f (x) is continuous and its derivative is piecewise continuous within
the periodic region. Fourier series of f (x) converges at speed of
1/n2.

References
Fourier series of full wave rectiﬁer

References
Complex Fourier series
Using Euler’s formula, Eq. 4 becomes
f (x) = a0 +
∞
k=1
ak − ibk
2
eikx
+
ak + ibk
2
e−ikx
Let c0 ≡ a0, ck ≡ ak −ibk
2 and c−k ≡ ak +ibk
2 , we have
f (x) =
∞
m=−∞
cmeimx
cm =
1
2π
π
−π
f (x)e−imx
dx (12)
eimx and einx are also mutually orthogonal provided n = m and it
forms a complete set. Therfore, it can be used as orthogonal basis.

References
Complex Fourier series
If f (x) has T period instead of 2π, x is replaced with 2πx/T.
f (x) =
∞
m=−∞
cmei 2πmx
T
cm =
1
T
T
2
−T
2
f (x)e−i 2πmx
T dx, m = 0, 1, 2... (13)

References
Fourier transform
from Eq. 13, we deﬁne variables k ≡ 2πm
T , ˆf (k) ≡ cmT√
2π
and
k ≡ 2π(m+1)
T − 2πm
T = 2π
T .
We can have
f (x) =
1
√
2π
∞
m=−∞
ˆf (k)eikx
k
ˆf (k) =
1
√
2π
T
2
−T
2
f (x)e−ikx
dx

References
Fourier transform
Let T −→ ∞
f (x) =
1
√
2π
∞
−∞
ˆf (k)eikx
dk (14)
ˆf (k) =
1
√
2π
∞
−∞
f (x)e−ikx
dx (15)
Eq.15 is the Fourier transform of f (x) and Eq.14 is the inverse
Fourier transform of ˆf (k).

References
Properties of Fourier transform
f (x), g(x) and h(x) are functions and their Fourier transforms are
ˆf (k), ˆg(k) and ˆh(k). a, b x0 and k0 are real numbers.
Linearity: If h(x) = af (x) + bg(x), then Fourier transform of
h(x) equals to ˆh(k) = aˆf (k) + bˆg(k).
Translation: If h(x) = f (x − x0), then ˆh(k) = ˆf (k)e−ikx0
Modulation: If h(x) = eik0x f (x), then ˆh(k) = ˆf (k − k0)
Scaling: If h(x) = f (ax), then ˆh(k) = 1
a
ˆf (k
a )
Conjugation: If h(x) = f ∗(x), then ˆh(k) = ˆf ∗(−k). With this
property, one can know that if f (x) is real and then
ˆf ∗(−k) = ˆf (k). One can also ﬁnd that if f (x) is real and then
|ˆf (k)| = |ˆf (−k)|.

References
Properties of Fourier transform
If f (x) is even, then ˆf (−k) = ˆf (k).
If f (x) is odd, then ˆf (−k) = −ˆf (k).
If f (x) is real and even, then ˆf (k) is real and even.
If f (x) is real and odd, then ˆf (k) is imaginary and odd.
If f (x) is imaginary and even, then ˆf (k) is imaginary and even.
If f (x) is imaginary and odd, then ˆf (k) is real and odd.

References
Dirac delta function
Dirac delta function is a generalized function deﬁned as the
following equation.
f (0) =
∞
−∞
f (x)δ(x)dx
∞
−∞
δ(x)dx = 1 (16)
The Dirac delta function can be loosely thought as a function
which equals to inﬁnite at x = 0 and to zero else where.
δ(x) =
+∞, x = 0
0, x = 0

References
Dirac delta function
From Eq.15 and Eq.14
ˆf (k) =
1
2π
∞
−∞
∞
−∞
ˆf (k )eik x
dk e−ikx
dx
=
1
2π
∞
−∞
∞
−∞
ˆf (k )ei(k −k)x
dxdk
Comparing to ”Dirac delta function”, we have
ˆf (k) =
∞
−∞
ˆf (k )δ(k − k)dk
δ(k − k) =
1
2π
∞
−∞
ei(k −k)x
dx (17)
Eq.17 doesn’t converge by itself, it is only well deﬁned as part of
an integrand.

References
Convolution theory
Considering two functions f (x) and g(x) with their Fourier
transform F(k) and G(k). We deﬁne an operation
f ∗ g =
∞
−∞
g(y)f (x − y)dy (18)
as the convolution of the two functions f (x) and g(x) over the
interval {−∞ ∼ ∞}. It satisﬁes the following relation:
f ∗ g =
∞
−∞
F(k)G(k)eikx
dt (19)
Let h(x) be f ∗ g and ˆh(k) be the Fourier transform of h(x), we
have
ˆh(k) =
√
2πF(k)G(k) (20)

References
Parseval relation
∞
−∞
f (x)g(x)∗
dx =
∞
−∞
1
√
2π
∞
−∞
F(k)eikx
dk
1
√
2π
∞
−∞
G∗
(k )e−ik x
dk dx
=
∞
−∞
1
2π
∞
−∞
F(k)G∗
(k )ei(k−k )x
dkdk
By using Eq. 17, we have the Parseval’s relation.
∞
−∞
f (x)g∗
(x)dx =
∞
−∞
F(k)G∗
(k)dk (21)
Calculating inner product of two fuctions gets same result as the
inner product of their Fourier transform.

References
Cross-correlation
Considering two functions f (x) and g(x) with their Fourier
transform F(k) and G(k). We deﬁne cross-correlation as
(f g)(x) =
∞
−∞
f ∗
(x + y)g(x)dy (22)
as the cross-correlation of the two functions f (x) and g(x) over
the interval {−∞ ∼ ∞}. It satisﬁes the following relation: Let
h(x) be f g and ˆh(k) be the Fourier transform of h(x), we have
ˆh(k) =
√
2πF∗
(k)G(k) (23)
Autocorrelation is the cross-correlation of the signal with itself.
(f f )(x) =
∞
−∞
f ∗
(x + y)f (x)dy (24)

References
Uncertainty principle
One important properties of Fourier transform is the uncertainty
principle. It states that the more concentrated f (x) is, the more
spread its Fourier transform ˆf (k) is.
Without loss of generality, we consider f (x) as a normalized
function which means
∞
−∞ |f (x)|2dx = 1, we have uncertainty
relation:
∞
−∞
(x − x0)2
|f (x)|2
dx
∞
−∞
(k − k0)2
|ˆf (k)|2
dk
1
16π2
(25)
for any x0 and k0 ∈ R. [3]

References
Fourier transform of a Gaussian pulse
f (x) = f0e
−x2
2σ2 eik0x
ˆf (k) =
1
√
2π
∞
−∞
f (x)e−ikx
dx
=
f0
1/σ2
e
−(k0−k)2
2/σ2
|ˆf (k)|2
∝ e
−(k0−k)2
1/σ2
Wider the f (x) spread, the more concentrated ˆf (k) is.

References
Signals with diﬀerent width.

References
The bandwidth of the signals are diﬀerent as well.

References
Nyquist critical frequency
Critical sampling of a sine wave is two sample points per cycle.
This leads to Nyquist critical frequency fc.
fc =
1
2∆
(26)
In above equation, ∆ is the sampling interval.
Sampling theorem: If a continuos signal h(t) sampled with interval
∆ happens to be bandwidth limited to frequencies smaller than fc.
h(t) is completely determined by its samples hn. In fact, h(t) is
given by
h(t) = ∆
∞
n=−∞
hn
sin[2πfc(t − n∆)]
π(t − n∆)
(27)
It’s known as Whittaker - Shannon interpolation formula.

References
Signal h(t) is sampled with N consecutive values and sampling
interval ∆. We have hk ≡ h(tk) and tk ≡ k ∗ ∆,
k = 0, 1, 2, ..., N − 1.
With N discrete input, we evidently can only output independent
values no more than N. Therefore, we seek for frequencies with
values
fn ≡
n
N∆
, n = −
N
2
, ...,
N
2
(28)

References
Fourier transform of Signal h(t) is H(f ). We have discrete Fourier
transform Hn.
H(fn) =
∞
−∞
h(t)e−i2πfnt
dt ≈ ∆
N−1
k=0
hke−i2πfntk
= ∆
N−1
k=0
hke−i2πkn/N
Hn ≡
N−1
k=0
hke−i2πkn/N
(29)
Inverse Fourier transform is
hk ≡
1
N
N−1
n=0
Hnei2πkn/N
(30)

References
Periodicity of discrete Fourier transform
From Eq.29, if we substitute n with n + N, we have Hn = Hn+N.
Therefore, discrete Fourier transform has periodicity of N.
Hn+N =
N−1
k=0
hke−i2πk(n+N)/N
=
N−1
k=0
hke−i2πk(n)/N
e−i2πkN/N
= Hn (31)
Critical frequency fc corresponds to 1
2∆ .
We can see that discrete Fourier transform has fs period where
fs = 1/∆ = 2 ∗ fc is the sampling frequency.

References
Aliasing
If we have a signal with its bandlimit larger than fc, we have
following spectrum due to periodicity of DFT.
Aliased frequency is f − N ∗ fs where N is an integer.

References
Aliasing example: original signal
Let’s say we have a sinusoidal sig-
nal of frequency 0.05. The sampling interval is 1. We have the signal

References
Aliasing example: spectrum of original signal
and we have its spectrum

References
Aliasing example: critical sampling of original signal
The critical sampling interval of the original signal is 10 which is
half of the signal period.

References
Aliasing example: under sampling of original signal
If we sampled the original sinusiodal signal with period 12, aliasing
happens.

References
Aliasing example: DFT of under sampled signal
fc of downsampled signal is 1
2∗12, aliased frequency is
f − 2 ∗ fc = −0.03333 and it has symmetric spectrum due to real
signal.

References
Aliasing example: two frequency signal
Let’s say we have a signal containing two sinusoidal signal of
frequency 0.05 and 0.0125. The sampling interval is 1. We have
the signal

References
Aliasing example: spectrum of two frequency signal
and we have its spectrum

References
Aliasing example: downsampled two frequency signal
Doing same undersampling with interval 12.

References
Aliasing example: DFT of downsampled signal
We have the spectrum of downsampled signal.

References
Filtering
We now want to get one of the two frequency out of the signal.We
will adapt a proper rectangular window to the spectrum.
Assuming we have a ﬁlter function w(f ) and a multi-frequency
signal f (t), we simply do following steps to get the frequency band
we want.
F−1
{w(f )F{f (t)}} (32)

References
Filtering example: ﬁltering window and signal spectrum

References
Filtering example: ﬁltered signal

References
References
Supplementary Notes of General Physics by Jyhpyng Wang,
http:
//idv.sinica.edu.tw/jwang/SNGP/SNGP20090621.pdf
http://en.wikipedia.org/wiki/Fourier_series
http://en.wikipedia.org/wiki/Fourier_transform
http://en.wikipedia.org/wiki/Aliasing
http://en.wikipedia.org/wiki/Nyquist-Shannon_
sampling_theorem
MATHEMATICAL METHODS FOR PHYSICISTS by George
B. Arfken and Hans J. Weber. ISBN-13: 978-0120598762
Numerical Recipes 3rd Edition: The Art of Scientiﬁc
Computing by William H. Press (Author), Saul A. Teukolsky.
ISBN-13: 978-0521880688Zong-han, Xieicbm0926@gmail.com Introduction to Fourier transform and signal analysis

References
References
Chapter 12 and 13 in
http://www.nrbook.com/a/bookcpdf.php
http://docs.scipy.org/doc/scipy-0.14.0/reference/ﬀtpack.html
http://docs.scipy.org/doc/scipy-0.14.0/reference/signal.html

Introduction to Fourier transform and signal analysis

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Introduction to Fourier transform and signal analysis (20)

Recently uploaded (20)

Introduction to Fourier transform and signal analysis