Lect4&5

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Institute of Integrated Information Systems

3. TIME & FREQUENCY (FOURIER ANALYSIS)

Specifies the relationships between waveforms and spectra, i.e. time and frequency

domains.Applies also to linear time-invariant (LTI) systems.

Fourier analysis technique used depends on whether signals being considered are periodic

or non-periodic.

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3.1 Periodic Signals & the Fourier SeriesWhen signal to be analysed is periodic in time, its spectrum can be evaluated using a

Fourier Series in either (a) trigonometric or (b) complex exponential form.

(i) Trigonometric Fourier Series

Consider arbitrary periodic waveform:

Where 1 = fundamental angular frequency off(t). f(t) can be described by:

f t A

A n t

B n t

n

n

n

n

( )

cos ( )

sin ( )

= +

+=

=

0

1

1

1

1

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Institute of Integrated Information systems

Procedure for Computing A0

A0represents the zero frequency, DC or mean level of the signal over one period. It is

computed as follows by integrating both sides of (1) over one period:

Interchanging and gives:

Terms (ii) and (iii) are zero for all n since they represent integration of a sinusoid over an

integral number of periods. Thus:

or:

f t dt A dt A n t dt B n t dt o n nnn

T

T

T

T

T

T

T

T

( ) cos( ) sin( )= + +=

=

1 111

2

2

2

2

2

2

2

2

f t dt A T A n t dt B n t dt o n nnn

T

T

T

T

T

T

( ) cos( ) sin( )= + + =

=

1 1

112

2

2

2

2

2

(i) (ii) (iii)

A T f t dt o T

T

= ( )2

2

AT

f t dt o T

T

=

1

2

2

( )

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Procedure for Computing An and Bn

Multiply both sides of (1) by cos(m1t) and integrate over one period:

For all mn, term (ii) is zero; for all m, term (iii) is zero. Hence, need only consider (ii) for

m=n.

Hence:

for all n.Similarly, by multiplying both sides of (1) by sin(m1t):

f t m t dt A m t dt A n t m t dt

B n t m t dt

T

T

T

T

T

T

T

T

o nn

n

n

i ii

iii

( ) cos( ) cos( ) cos( ) cos( )

sin( ) cos( )

( ) ( )

( )

1 1 1 1

1

1 1

1

2

2

2

2

2

2

2

2

= + +

=

=

[ ]

f t n t dt n t dt

An t dt

A T

T

T

T

T

T

T

n

n

( ) cos( ) cos ( )

cos( )

1

2

1

1

2

2

2

2

2

2

21 2

2

=

= +

=

A T f t n t dt n T

T

= 2

12

2

( ) cos ( )

BT

f t n t dt n T

T

=

21

2

2

( ) sin ( )

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Interpretation of A0, An and Bn

A0is the DC component, or mean value, off(t).

An and Bn are the magnitudes of the cos(n1t) and sin(n1t) components,respectively, for each value ofn.

Any spectrum, and hence waveform, is defined completely by the set of

values forA0,An and Bn.

The integral of the product in:

has the same form as a correlation integral. It is therefore a measure of the

similarity between f(t) and cos(n1t).

AT

f t n t dt n T

T

=

21

2

2

( ) cos ( )

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Orthogonal Components

As seen previously:

for all mn, and

for all mn, and

for all m, where m and n are integers and 1 = 2/T.

In the above, the two terms within the integration are orthogonal over period T.

sin( ) sin( )m t n t dt T 1 1 0=

cos( ) cos( )m t n t dt T

1 1 0=

sin( ) cos( )m t n t dt T

1 1 0=

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Implications of Orthogonality

In general, Fourier Series for:

Average power series due to f(t) in a 1 load:

If Fourier Series substituted in (2), the following terms result from squaring:

Plus cross terms such as:

Because of orthogonality, cross terms integrate to zero over one period and:

f t A A t B t

A t B t

A t B t

( ) cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

= + + +

+ ++ +

0 1 1 1 1

2 1 2 1

3 1 3 1

2 2

3 3

[ ]PT

f t dt ave T

T

=

12

2

2

2

( ) ( )

A A t B t A t B t

A k t B k t k k

0

2

1

2 2

1 1

2 2

1 2

2 2

1 2

2 2

12 2

1

2 2

1

2 2, cos ( ) , sin ( ) , cos ( ) , sin ( ) ,

, cos ( ) , sin ( ) ,

A A t A t A t A t B t etc0 1 1 1 1 2 1 1 1 1 12cos( ) , cos( ) cos( ) , cos( ) sin( ) , .

PT

A dt T

A n t dt T

B n t dt ave nn

n

nT

T

T

T

T

T

= + +

=

=

1 1 10

2 2 2

1

1

2 2

1

12

2

2

2

2

2

cos ( ) sin ( )

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(ii) Complex Exponential Fourier Series

Sometimes a simpler version to use than trigonometric.

Note:

where . Hence, the information in the trigonometric series can, in principle, be

represented in complex exponential form. For any n:

Hence, the Fourier series forf(t) is:

or:

Equation (2) defines the complex exponential Fourier Series.

Cn are complex coefficients.

Note that at n=0, C0=A0.

e n t n t n tj j

1

1 1= +cos( ) sin( )

j = 1

[ ] [ ] A n t B n t A

e eB

e e

A Be

A Be

C e C e

n nn n t n t n n t n t

n n n t n n n t

nn t

nn t

cos( ) sin( )

( )

1 12 2

2 2

1 1 1 1

1 1

1 1

+ = + +

=

+

+

= +

j j j j

j -j

j j

jj j

[ ] f t A C e C enn t

n

n t

n

( )( )= + +

=

01

1 1j j

)2()( 1j

=

=n

tn

n eCtf

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(ii) Complex Exponential Fourier Series (contd.)

Also, for positive n:

which using trigonometric Fourier Series coefficients:

For negative n:

And generally:

CA B

n

n n=

j

2

[ ]=

=

1

2

2

1

1 12

2

1

2

2

T f t n t n t dt

T f t e dt

T

T

T

Tn t

( ) cos( ) sin( )

( )

j

j

[ ]

CA B

T f t n t n t dt

T f t e dt

n

n n

n t

T

T

T

T

=+

= +

=

j

j

j(-

21

2

2

1

1 12

2

1

2

2

( ) cos( ) sin( )

( )

)

CT

f t e dt

f t C e

n

n t

n

n t

n

T

T

=

=

=

11

2

2

1

( )

( )

j

j

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3.2 Symmetry & the Fourier Series (contd.)

Half-Wave or Inverse-Repeat (IR)

Odd harmonics are IR (both sines and cosines).

AllAn and Bn are zero for even n;

all Cn are zero for even n.

Note: a square wave has both IR and odd symmetry.

AllAn are zero;

all Bn are zero for even n;

all Cn are imaginary and zero for even n.

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3.3 Negative FrequenciesIn the complex exponential Fourier Series, n can be negative, indicating negative

frequencies. What does negative frequency mean?

Methods of producing a real sinewave from rotating vectors:

(a)

(b)

(a) corresponds to trigonometric Fourier Series model;

(b) corresponds to complex exponential Fourier Series model.

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3.4 Derivation of Fourier Transform from Fourier SeriesThe Fourier Transform allows the spectrum of a non-periodic signal to be computed.

Simplest approach is to let T for a non-periodic signal.

Periodic Spectrum (Line Spectrum)

Non-Periodic Spectrum (Continuous Spectrum)

As T, spectral lines merge to form a continuous spectrum.

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3.4 Derivation of Fourier Transform from Fourier Series (contd.)

Start with the complex exponential Fourier Series:

f t C en n tn

( ) ==

j

1 CT

f t e dt n jn t T

T

= 11

2

2

( )

For a non periodic signalT

and C Hence define F C T

n

n n

=

:

0

f tF

Ten n t

n

( ) ==

j 1 F f t e dt n n t=

( )j 1

j jn 1

F Fn ( )j

f t F e d t( ) ( )=

1

2

j j F f t e dt t( ( )j ) =

j

-

f t F ( ) ( ) j

1

12 1

2= =T T

TT

d

1

2

n

T

( ) is a continuous frequency variable

These are defining s for the Fourier Transform abbreviated toexpression ; :

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3.5 The Discrete Fourier TransformFor the analysis of the spectrum of an analogue signal by a digital computational procedure.

Notes:

(i) Essentially treats a non-periodic waveform as if it were periodic (more like Fourier Series).

(ii) Produces both magnitude and phase spectra at discrete frequency values.

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