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