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Frequency Domain Analysis of Signals

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Frequency Domain Analysis of Signals

The spectral representation oI signal basically involve the decomposition oI the
signal in terms oI sinusoidal (or complex exponential) components. With such a
representation, a signal is said to be in the Irequency domain. The recombination oI
the sinusoidal components to reconstruct the original signal is called spectral
synthesis. The physical analogy oI spectral analysis and synthesis is shown in
Eigure 1.

Eigure 1. (a) Analysis and (b) synthesis oI white light (sunlight) using glass prisms.

The Concept of Frequency in Continuous Time and Discrete Time
The concept oI Irequency is directly related to the concept oI time. Actually it
has the dimension oI inverse time. Thus we should expect that the nature oI time
(continuous or discrete) would aIIect the nature oI the Irequency accordingly
Continuous-Time Sinusoidal Signals
∞ < < ∞ − + = + Ω = t Ft A t A t x

), 2 cos( ) cos( ) ( θ π θ (1)
where Ω (rad/s) is called angular Irequency and E (cycle/s,Hertz) is called cyclic
Irequency. The analog sinusoidal signal is characterized by the Iollowing properties.
A1) Eor every Iixed value oI E, x
a
(t) is periodic, i.e. x
a
( t ¹ T
p
) ÷ x
a
(t).
A2) Continuous sinusoidal signals with distinct Irequencies are themselves distinct.
A3) Increasing the Irequency E results in an increase in the rate oI oscillation oI the
signal. Due to continuity oI the time variable t, we can increase the Irequency, E,
without limit, with a corresponding increase in the rate oI oscillation.
By deIinition, Irequency is an inherently positive physical quantity (i.e. number
oI cycles per unit time). However, in many cases, only Ior mathematical
convenience, we need to introduce negative Irequency. To see this we recall that
) ( ) (
2 2
) cos( ) (
θ θ
θ
+ Ω − + Ω
+ = + Ω =


e
A
e
A
t A t x (2)
As time progress the phasors rotate in opposite direction with angular Irequency ± Ω.
Eor mathematical convenience, we use both negative and positive Irequencies.
Hence, the Irequency range Ior analog sinusoidal is -∞ · E · ∞.

Discrete-Time Sinusoidal Signals
∞ < < ∞ − + = + = n fn A n A n x ), 2 cos( ) cos( ) ( θ π θ ω (3)
ω is the Irequency in radians per sample, and I is the Irequency in cycles per sample.
In contrast to continuous-time sinusoidal, the discrete time sinusoidals are
characterized by the Iollowing properties.
B1) A discrete-time sinusoid is periodic only iI its Irequency I is a rational number.
ProoI: iI x is periodic, then x(n¹N) ÷ x(n) Ior all n,
Cos(2πI
0
(N¹n)¹θ) ÷ Cos(2πI
0
n¹θ)
! 2πI
0
N÷2kπ, where k is integer
∴ I
0
÷k/N
B2) Discrete-time sinusoids whose Irequency are separated by an integer multiple oI
2π are identical.
ProoI: cos|(ω
0
¹2π)n¹θ| ÷ cos(ω
0
n¹2πn¹θ) ÷ cos(ω
0
n¹θ)
As a result, x
k
(n) ÷ A cos(ω
k
n¹θ), ω
k
÷ ω
0
¹2kπ, k ÷ 0,1,2, .
Are indistinguishable. On the other hand, the sequences oI any two sinusoids
with Irequencies in the range -π · ω · π or 1/2 · I · 1/2 are distinct.
ThereIore, sinusoidal signal with Irequency ,ω, ≤ π or ,I, ≤ 1/2 are unique. We
call the sinusoid having the Irequency ,ω, ~ π or ,I, ~ 1/2 an alias oI a
corresponding sinusoid with Irequency ,ω, ≤ π or ,I, ≤ 1/2.
B3) The highest rate oI oscillation in a discrete-time sinusoid is attained when ,ω, ÷ π
(or ,E, · 1/2).

Effect of Sampling on Frequency
The most oIten used uniIorm sampling is described by the relation
) ( ) (

nT x n x = (4)
The time interval T
s
between successive samples is called the sampling period or
sample interval and its reciprocal 1/T
s
÷ E
s
is called the sampling rate or the sampling
Irequency.
Periodic sampling establishes a relationship between the time variables t and n oI
continuous-time and discrete-time signals, respectively. Indeed, these variables are
linearly related through the sampling period T
s
or, equivalently, through the sampling
rate E
s
÷ 1/T
s
, as


F
n
nT t = = (5)
As a consequence oI Eqn. (5), there exists a relationship between the Irequency
variable E (or Ω) Ior analog signals and Irequency variable I (or ω) Ior discrete-time
signals. To establish this relationship, consider an analog sinusoidal signal oI the
Iorm oI Eqn. (1), which, when sampled periodically at a rate E
s
÷1/T
s
samples per
second, yields
)
2
cos( ) 2 cos( ) ( ) ( θ
π
θ π + = + = ≡


F
nF
A FnT A n x nT x (6)
II we compare Eqn. (3) with Eqn. (6), we note tat the Irequency variables E and I are
linearly related as


F
F
f = (7a)
or, equivalently, as


T Ω = ω (7b)
The relation in Eqn. (7) justiIies the name relative or normalized Irequency, which is
sometimes used to describe the Irequency variable, I. As Eqn. (7) implies, we can
use I to determine the Irequency E in hertz only iI the sampling Irequency E
s
is known.
Recall Irom the previous section, the relations among Irequency variables are
summarized in Table 1.

Table 1 Relations among Irequency variables
Continuous-time signals Discrete-time Signal
Ω ÷ 2 π E ω ÷ 2 π I
Radians/sec Hz

Radians/sample Cycles/Sample
-∞ · Ω · ∞ ω ÷ ΩT
s
, I ÷ E/E
s
-π · ω · π
-∞ · E · ∞ Ω ÷ ω/T
s
, E ÷ I E
s
-1/2· I · 1/2
-π/T
s
≤ Ω ≤ π/T
s

-E
s
/2 ≤ E ≤ E
s
/2

Erom these relations we observe that the Iundamental diIIerence between
continuous-time and discrete-time signals is in their range oI values oI the Irequency
variables E and I, or Ω and ω. Periodic sampling oI a continuous-time signal implies
a mapping oI inIinite Irequency range Ior the variable E (or Ω) into a Iinite Irequency
range Ior the variable I (or ω). Since the highest Irequency in a discrete-time signal
is ω ÷ π or I ÷1/2, it Iollows that, with a sampling rate E
s
, the corresponding highest
values oI E and Ω are


T
F
F
2
1
2
max
= = (8a)


T
F
π
π = = Ω
max
(8b)
ThereIore, sampling introduces an ambiguity, since the highest Irequency in a
continuous-time signal that can be uniquely distinguished when such a signal is
sampled at a rate E
s
÷1/ T
s
is E
max
÷E
s
/2, or Ω
max
÷ πE
s
. To see what happens to
Irequencies above E
s
/2, let us consider Eigure 2 and an example in Eigure 3.


Eigure 2. Relationship between the continuous-time and discrete-time Irequency
variables in the case oI periodic sampling.


Eigure 3. Illustration oI aliasing
The Sampling Theorem
Given any analog signal, how should we select the sampling period, T
s
or,
equivalently, the sampling rate E
s
? To answer this question, we must have some
inIormation about the characteristics oI the signal to be sampled. In particular, we
must have some general inIormation concerning the Irequency content oI the signal.
We know that the highest Irequency in an analog signal that can be unambiguously
reconstructed when the signal is sampled at a rate E
s
is E
s
/2. Any Irequency above
E
s
/2 or below E
s
/2 results in samples that are identical with a corresponding
Irequency in the range E
s
/2 ≤ E ≤ E
s
/2. E
s
/2 is called the Iolding Irequency or
Nyquist Irequency. Suppose that the Irequencies do not exceed some known
Irequency, say E
max
, we must select the sampling rate to be suIIiciently high. That is,
we must select E
s
/2 to be greater than E
max
. Thus to avoid the problem oI aliasing, E
s

is selected so that
max
2F F

> (9)

Types of Signal (revisited)
The mathematical methods employed in the analysis oI discrete-time signals
depend on the characteristics oI the signals. The characteristics which relate to the
spectral analysis are discussed below.

Periodic Signal vs. Aperiodic Signal
Periodic signal: signal which is repetitive with a certain period D, such that
x(t) ÷ x(t¹D). Aperiodic signal: signal which is not periodic.

Energv Signal vs. Power Signal
The energy oI a signal is deIined as


+∞
∞ −
= dt t x E
2
) ( (Continuous) (10a)


+∞
−∞ =
=

n x E
2
) ( (Discrete) (10b)
II E is Iinite (i.e. E ≤ ∞ ), the signal is an energy signal. Energy signal is suIIicient to
guarantee the existence oI Eourier TransIorm. Most aperiodic signals oI practical
interest are energy signal.
The power oI a signal is deIined as


− ∞ →
=


dt t x
T
P
2
) (
2
1
lim (Continuous) (11a)


− =
∞ →
+
=



n x
N
P
2
) (
1 2
1
lim (Discrete) (11b)
II P is Iinite (i.e. P ≤ ∞ ), the signal is an power signal. Power signal is suIIicient to
guarantee the existence oI Eourier Series expansion. Most periodic signals oI
practical interest are power signal.

Spectral Analysis and Synthesis of Signals
Orthogonal Expansion
Eigenvalue Problems ! Orthogonal Vectors ! Eigenvector expansion
Sturm-Liouville Problems ! Orthogonal Eunctions ! EigenIunction expansion

Fourier Expansion (Spectral Expansion)
Eourier Series (Ior power signal) and Eourier TransIorm (Ior energy signal) are
basically orthogonal expansion oI a continuous Iunction using sinusoidal Iunctions or
complex exponential Iunctions. The signal is viewed as a superposition oI may
inIinitely long wave trains oI diIIerent Irequencies. The term spectrum is used when
reIerring to the Irequency content oI a signal. The spectrum provides an 'identity¨
or a signature Ior the signal in the sense that no other signal has the same spectrum.
This attribute is related to the mathematical treatments oI Irequency domain
technique.

Sampling in Time Domain
The continuous Eourier TransIorm is a powerIul technique but has the drawback that
the Iunctions (signals) must be known analytically over the complete domain. This
occurs in only rare cases making it unsuitable Ior practical situations, especially iI the
signals are experimental in origin. This problem may be solved by discretizing the
signal 'careIully¨ in time domain. II we consider both continuous time and discrete
time, this type oI orthogonal expansion can be summarized by Eigure 4. The Eourier
analysis and synthesis Iormulas are summarized in Eigure 5.

Power Signal ! Eourier Series
Continuous Time Signal
Energy Signal ! Eourier TransIorm
Power Signal ! Discrete Time Eourier Series
Discrete Time Signal
Energy Signal ! Discrete Time Eourier TransIorm
Sampling Theorv

Eigure 4. Summary oI Iourier expansions Ior signals

Sampling in Frequencv Domain
In spectral analysis, the behavior oI a signal is viewed as a superposition oI may
inIinitely long wave trains oI diIIerent Irequencies. The actually response is
synthesized by a judicious combination oI these wave trains. Thus the problem oI
characterizing a signal is transIormed into one oI determining the set oI combination
coeIIicients. These coeIIicients are called the Eourier transIorm oI the signal.
While the problem being tackled invariantly simpliIies when it is expressed in terms
oI the Eourier transIorm, the last step in the analysis involves perIorming an inverse
transIorm (reconstructing the signal) and this, generally, is very diIIicult to do in an
exact analytical manner. Analytical transIorms are Ieasible only iI the Iunction to be
transIormed is mathematically simple. UnIortunately, this is not the case in any
situation oI practical interest. This problem may be solved by discretizing the
spectrum 'careIully¨ in Irequency domain. When both the signal and the spectrum
are approximated by discretization, the spectral analysis is called Discrete Eourier
TransIorm (DET). To summarize, the Iormulas Ior the DET and IDET (Inverse
Discrete Eourier TransIorm) are
DET


=

− = =
1
0
/ 2
1 , , 2 , 1 , 0 , ) ( ) (



N k e n x k X !
π
(12a)
IDET


=
− = =
1
0
/ 2
1 , , 2 , 1 , 0 , ) (
1
) (



N n e k X
N
n x !
π
(12b)

Eqn. (12) has an enormous advantage in that the East Eourier TransIorm (EET)
computer algorithm can be used Ior economically computing the transIormations.

Domain Aliasing
Discretization in time domain may cause Irequency aliasing as shown in Eigure 6.
Similarly, discretization in Irequency domain may cause time aliasing. The catch oI
eIIicient EET is that we need to careIully select sampling intervals to avoid or
minimize the aliasing problem.

Eigure 5. Summary oI analysis and synthesis Iormulas




6
Properties of the Discrete Fourier Transform
Properties oI DTET are shown in Table 1 and 2. Properties oI DET are shown in
Table 3 and 4. Some oI the DET properties are discussed below.

Svmmetrv
II the sequence x(n) is real, X(-k) ÷ X*(k). In this case we say that the spectrum oI a
real signal has Hermitian symmetry.

Periodicitv
II x(n) and X(k) are an N-point DET pair, then
x(n¹N) ÷ x(n) Ior all n
X(k¹N) ÷ X(k) Ior all k

Dualitv
One oI the most important properties oI the discrete Eourier transIorm is the duality
relation between the time domain and the Irequency domain, where periodicity with
'period¨ α in one domain implies discretization with 'spacing¨ oI 1/α in the other
domain, and vice versa as shown in Eigure 7. The amount oI inIormation contained
in one domain is limited by the length oI the window oI that domain (i.e. the period α).
ThereIore, the sampling process reduces the inIormation contained in a
continuous-time signal. II the spectrum oI the analog signal can be recovered Irom
the spectrum oI the discrete-time signal, there is no loss oI inIormation. The
criterion oI selecting a sampling rate is dictated by the sampling theorem.
Linearitv
II



and



Then
x
1
(n)
DET
N
X
1
(k)
x
2
(n)
DET
N
X
2
(k)



X(n∆t)
t
0
T÷1/∆f
∆t
f
0
∆f
X(n∆f)
Time
Window
Erequency
Window
F÷1/∆t

Eigure 7 Duality relation between time domain and Irequency domain

Time and Frequencv Shift
II



Then






Energv
The energy in a signal x(n) can be computed as the summation oI the square oI the
amplitude oI the signal at each point. Each Eourier coeIIicient X(k) indicates the
amplitude oI the sinusoid oI Irequency ω
k
÷ 2πk/N that is contained in the
'synthesized¨ signal. ThereIore, the energy in the signal can be computed both in
time domain and Irequency domain as described by Parseval`s identity
∑ ∑

=

=
=
1
0
2
1
0
2
) (
1
) (




k X
N
n x (13)
The plot oI ,X(k),
2
versus Irequency is the auto-spectral density oI the signal (also
known as power spectral density).
a
1
x
1
(n)¹a
2
x
2
(n)
DET
N
a
1
X
1
(k)¹a
2
X
2
(k)
x (n)
DET
N
X (k)
x(n-m)
DET
N
X(k)e
-j2πkm/N
X(n-m)
DET
N
x(n)e
-j2πkm/N


2
1






3
4
Computation of Fourier Transform - Fast Fourier Transform
The East Eourier TransIorm (EET) is an algorithm Ior computing the Discrete Eourier
TransIorm (DET). BeIore the development oI the EET the DET required excessive
amounts oI computation time, particularly when high resolution was required (large
N). The EET Iorces one Iurther assumption, that N is a multiple oI 2. This allows
certain symmetries to occur reducing the number oI calculations (spediIically
multiplication) which have to be done. The EET algorithm is one kind oI
divide-and-conquer algorithm.

Plots
A signal in the time domain is primarily plotted as x(n) vs. time n. However, there
are several appealing alternatives in the Irequency domain that can Iacilitate the
interpretation oI the inIormation encoded in the signal. Consider the signal shown in
Eigure 8, obtained Irom the Iree vibration oI a column aIter it is Ireed Irom an initial
static displacement.

• Eigure 8b shows the auto-spectral density versus the Irequency index k. The
Iirst mode oI vibration is clearly seen. The auto-spectral density can be plotted
in log scale to highlight other vibration modes oI lower amplitude. II a process
causes an exponential change in amplitude with Irequency, the semi-log scale
renders a linear trend.
• Eigures 8c and 8d show Re|X(k)| and Im|X(k)| versus the Irequency index k.
• Eigure 8e and 8I show the amplitude and the phase versus the Irequency index
k.
• Eigure 8g shows the imaginary component versus the real component and real
component. This is called the Cole-Cole plot and it can be used to identiIy
materials that show relaxation behavior (e.g., response oI a viscoelastic
material): a relaxation deIines a semi-circle.
• Eigure 8h shows a plot oI amplitude vs. phase.






8
Practical Considerations
Frequencv Resolution -- Zero-padding
The relationship between the time resolution and Irequency resolution is
t f
N
∆ ∆
=
1

This is known as the 'uncertainty principle¨ in signal processing. It prescribes that
given N pieces oI inIormation, the resolution in Irequency can only be improved at
the expense oI the resolution in time. This is an inherent trade-oII in discrete signal
processing.
A signal can be extended in the time domain by adding zeros at the tail to increase N.
This practice is known as 'zero-padding.¨ It is implemented aIter acquisition,
during processing. Zero-padding does not aIIect the total Irequency width. An
example is shown in Eigure 9.


9
Truncation Leakage Windowing


10

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