Signals and Systems 1 -...

Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 1

FachgebietNachrichtentechnische Systeme

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Signals and Systems 1

Prof. Dr.-Ing. I.Willms

(Version 2.1)



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Contents

1 Introduction

2 Signal representations in the time- and frequency

domain

3 Analog systems

4 Discrete systems



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1 IntroductionSignals and Systems

• Signals represent a physical quantity changing over time • Signal usually contain some information relevant for the

observer of the signal• Signals exhibit totally different dimension depending on

the application• Signals can be defined mathematically with/without

physical counterparts

• Systems exhibit an input and and output• Typically systems have a certain task (signal processing)• Output signal is a function (transform) of the input signal



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1 IntroductionEssential tasks in communication engineering

Ia Transmission of analog/digital baseband signals• Output y(t) should follow/be identical to input signal s(t)

- regardless of noise in communication channel- regardless of transfer characteristics of channel

• Example: Transmission of video/audio signals over a long cable

Ib Transmission of analog/digital signal by means of a carrier• Additional Modulation/demodulation is required• Reason: Inefficient/impossible base band communication• Examples:

– Transmission of video/audio signals via satellite using MW signals– Communication via mobile phones or cordless phones



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1 IntroductionEssential tasks in communication engineering

IIa Detection of a known signal in the presence of noise• Examples: • Switching on the lights/activation of apparatus

(like door openers by means of wireless remote control• Detection of an intrusion by means of detectors• Access control

IIb Estimation of signal parameters in the presence of noise• Automatic collision control by means of determining distance to others cars• Determination of air velocity by means of US time-of-flight methods



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2.1 Analog, Discrete and Digital Signals

s

t0

s

0k od. k t

Analog signal sequence

s

0

signal with discrete values

s

k od. k t

Digital signal

continuoustime-(space-) domain

discrete

t0

Analog signal

Con

tinuo

usdi

scre

te

Ran

ge o

f val

ues



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2.2 Deterministic Signals in the Time Domain2.2.1 The Exponential Signal

( ) cos sinj ts t e t j tω ω ω= = +

( )ˆ ˆ ˆ( ) cos( ) Re Re where u uj t jj tuu t u t u e u e u u eω ϕ ϕωω ϕ += ⋅ + = ⋅ = ⋅ = ⋅

( )j t t j t pte e e eσ ω σ ω+ = ⋅ =

For voltages it holds:

For increasing/decreasing signals:



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2.2.2 The Exponential Sequence

( ) for ks k z k Z= ∈

( ) cos( ) sin( )j T ks k e Tk j Tkω ω ω⋅ ⋅= = + ⋅

( )( )

cos( ) sin( )

pTk j k T kT j kT

kT k T

s k e e e e

e Tk j e Tk

σ ω σ ω

σ σω ω

+ ⋅

⋅

= = = ⋅

= ⋅ + ⋅ ⋅



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2.2.3 The Dirac Function

0

1( ) limT

trectT T

δ τ→

⎛ ⎞= ⎜ ⎟⎝ ⎠ 0

( )δ τ

τ

Approximation:



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2.2.3 The Dirac Function

0 0( ) ( ) ( ) with ( ) as an arbitrary signalt t t t dt tδ+∞

−∞

Φ = − ⋅Φ Φ∫

1( ) ( )at ta

δ δ= ⋅

If 1 then: ( ) ( )a t tδ δ= − − =

( ) ( ) ( )s t t s dδ τ τ τ+∞

−∞

= − ⋅∫

Definition:

Properties:



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2.2.4 The Unit Impulse

0

1 for 0( ) ( )

0 for 0k

s k kk

γ=⎧

= = ⎨ ≠⎩

2− 20 5

1( )0 kγ

k



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2.2.5 The Step Function

0 for 0( )

1 for 0t

tt

ε<⎧

= ⎨ ≥⎩

( ) ( )t

t dε δ τ τ−∞

= ∫( )tε

t0

1



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2.2.6 The Step Sequence

1

0 for 0( )

1 for 0k

kk

γ −

<⎧= ⎨ ≥⎩

1( )kγ −

1

5− 2− 0 2 5 k



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2.2.7 Periodic Signals

( ) ( ) where ,..., 1, 1,...,s t s t nT n= + = −∞ − + +∞

2 1 0( ) ( )n

s t s t nT+∞

=−∞

= −∑

General property:

Transform of impulses into a periodic signal:

2 1 0( ) ( ) with as weighting factorsn nn

s t c s t nT c+∞

=−∞

= −∑



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2.2.7 Periodic Signals

1

02

0

( )

( )

n

n

ts t rectT

t nTs t rectT

Ttrect nT T

+∞

=−∞

+∞

=−∞

⎛ ⎞= ⎜ ⎟⎝ ⎠

−⎛ ⎞⇒ = ⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠

∑

∑

0n = 1n = 2n =

T

( )2s t

0T 02T t

Example:



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2.2.8 Impulse Type Signals

11 for 2( )10 for 2

xrect x

x

⎧ ≤⎪⎪= ⎨⎪ >⎪⎩

( )rect x

012

1

1

12

−1− x



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2( / )( ) t Ts t e−=( )s t

3− 2− 1− 32100

1

/t T

The Gaussian impulse



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1 for 0( ) ( ) 0 for 0

1 for 0

ts t sign t t

t

>⎧⎪= = =⎨⎪− <⎩

( )sign t

t

1

1−

0



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1 for ( )

0 otherwise

t t Tts t TT

⎧− ≤⎪⎛ ⎞= Λ = ⎨⎜ ⎟

⎝ ⎠ ⎪⎩

( ) ts tT

⎛ ⎞= Λ⎜ ⎟⎝ ⎠

TT−t

1



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2.2.8 Impulse Type Signals2

1

00 1( ) cos( ( ))

t tts t e t tω

⎛ ⎞−−⎜ ⎟⎝ ⎠= ⋅ −

( )s t

1/t t1− 0 1 2 3 4

0

1

The figure showss(t) for t1 = t0



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2.2.9 Adjustment of Time and FrequencyFunctions

2 1( ) ts t a sb

⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠

2 0( )2ts t u rectT

⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠

2 1( ) ( )s t s t Tν= −

Case1: Change of amplitude, compression & expansion with regard to time axis

Case2: Shift (Time delay or advance)

Example:



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1

2 0 1

0

0

( )

( )2

2

2

ts t rectT

ts t u s

tu rectTtu rectT

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠⎛ ⎞= ⋅ ⎜ ⎟⎝ ⎠

Example for expansion: ( )rect x

012

1

1

12

−1− x

( )2s t

0 T

0u

T−t



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2 1( ) ( )s t s t= −

1

2 1

( ) ( )( ) ( )

( )

s t ts t s t

t

ε

ε

== −= −

Case3: Mirroring (b = -1)

Example:

t

1

0

( )2s t



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1 2

3 2

( ) ( ) 3

( ) ( )3

t ts t rect s t rectT T

t Ts t s t T rectT

νν

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−⎛ ⎞= − = ⎜ ⎟⎝ ⎠

2 1( ) ( )s t s t Tν= −

3 2 2

1 1

( ) Replace in ( ) argument only by

and do the same in ( )

t ts t as s t tb btas T s tb ν

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞= −⎜ ⎟⎝ ⎠

Combination of expansion & shift:

Combination of shift & expansion:

( )3s t

vT

1

t

3T



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1 0( ) ; ; 2ts t rect a u bT⎛ ⎞= = =⎜ ⎟⎝ ⎠

2 ( ) t T Tts t rect rectT T T

ν ν−⎛ ⎞ ⎛ ⎞= = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3 0( )2

T Tt ts t arect u rectbT T T T

ν ν⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Example:

( )3s t

0u2T

2 vT t



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2 1( ) ( )s t s t= −

3 2 1 1( ) ( ) ( ( )) ( )s t s t T s t T s T tν ν ν= − = − − = −

4 1( ) ( )s t s t Tν= −

5 4 1 3( ) ( ) ( ) ( )s t s t s t T s tν= − = − − ≠

Mirroring & shifting:

New sequence: Shifting & mirroring:



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( )t tr tT T

ε⎛ ⎞ = ⋅⎜ ⎟⎝ ⎠

1( ) ts t rT⎛ ⎞= ⎜ ⎟⎝ ⎠

2 1( ) ( ) ts t s t rT−⎛ ⎞= − = ⎜ ⎟

⎝ ⎠

3 2( ) ( ) T ts t s t T rTν

ν−⎛ ⎞= − = ⎜ ⎟

⎝ ⎠

Example with a ramp function r(t):

vT

( )3s t

( )1s t( )2s t

t

t



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vt T± ±

4 1( ) ( ) t Ts t s t T rT

νν

−⎛ ⎞= − = ⎜ ⎟⎝ ⎠

5 4( ) ( ) t Ts t s t rT

ν− −⎛ ⎞= − = ⎜ ⎟⎝ ⎠

There are 4 cases:

( )5s t ( )4s t

vT− vT t



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1 2 3

1 2 3

( ) ( ) where ( ) ( ) ( ( ))f x f y y f xf x f f x

= =

⇒ =

01

1

( )f rect ω ωωω

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

All methods described above can be extended to frequency functions.

Example:

In general one function can be used as the argument of another function.



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2.2.10 Energy and Power of Signals

21 ( )elE u t dtR

∞

−∞

= ∫

2 ( )E s t dt∞

−∞

= ∫

21lim ( )2

T

TT

P s t dtT

+

→∞−

= ∫

0 E< < ∞

0 or P E< < ∞ →∞

Electrical Energy:

Signal Energy:

Condition for energy signals:

Condition for power signals:

Signal Power:



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2.2.10 Energy and Power of Signals

2lim ( )k K

k k KE s k

=+

→∞=−

= < ∞∑

21lim ( )2

k K

k k K

P s kK

=+

→∞=−

= < ∞∑

Conditions for discrete signals:



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[ ]

0 1 0 1 2 0 2 0 0

0 01

0 0 01

ˆ ˆ( ) cos(2 ) cos(2 2 ) ... where 2

ˆ cos( )

and due to cos( ) cos cos sin sin :

ˆ ˆcos( ) cos sin( )sin

n nn

n n n nn

s t s s f t s f t f

s s n t

x y x y x y

s s n t s n t

π ϕ π ϕ ω π

ω ϕ

ω ϕ ω ϕ

∞

=

∞

=

= + + + ⋅ + + =

= + +

+ = −

+ −=

∑

∑

2.3.1 Periodic Signals and the Fourier Series

( ) ( ) integer, , T Periods t s t kT k k= + −∞ < < ∞ =

00

1fT

=0

1 , 1nf n nT

= >

Properties of periodic signals:

Fourier series onset with 3 essential components:



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00ˆ ˆSetting cos , sin , , one obtains:

2n n n n n naa s b s sϕ ϕ= − = =

[ ]00 0

1

2 200

1

2 2

( ) cos( ) sin( ) Trigonometric form2

cos( ) Polar form2

ˆ where arctan and

n nn

n n nn

nn n n n

n

as t a n t b n t

a a b n t

b s a ba

ω ω

ω ϕ

ϕ

∞

=

∞

=

= + +

⎡ ⎤= + + +⎣ ⎦

= − = +

∑

∑

Please observe limited range of values for the arctan function!



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0

0

00

0

1 ( ) (this is the time averaged value of s(t))2

t T

t

as s t dtT

+

= = ∫

0

0

00

2 ( ) cos( )t T

nt

a s t n t dtT

ω+

= ∫

0

0

00

2 ( )sin( )t T

nt

b s t n t dtT

ω+

= ∫

Determination of Fourier coefficients:



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2.3.1 Periodic Signals and the Fourier SeriesThe exponential form

0 for 0 with 02

n nn

a jbc n b−= ≥ =

2n n

n na jbc c ∗

−

+= =

02 Re The amplitude of the cos( ) for 0n na c n t nω= ≥

02 Im The amplitude of the sin( ) for 0

2 Im for 0n n

n

b c n t n

c n

ω= − ≥

= + <

Definition:

Relation to trigonometric coefficients:



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00 0

1

2 2

( ) 2 cos( )

1 where and arctan2

jn tn n n

n n

nn n n n n

n

s t c e c c n t

bc a b ca

ω ω ϕ

ϕ

+∞ +∞

=−∞ =

= = + +

= + = − = ∠

∑ ∑

Periodic signals thus are represented by:

Please observe:

Complex coefficients represent pointers which are rotated byexponential function (clockwise rotating for positive n)



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Additional Fourier series properties:

Linearity

Time delay

Reversal


( )

0 0 0 0

0 0

0 0

0

0 0

0

0

0 00 0

0 00

0

1 1 for 02 21 2 2 ( ) cos( ) ( ) sin( )2 2

1 ( ) cos( ) sin( )

1 ( )

n n n

t T t T

t t

t T

t

t Tjn t

t

c a j b n

js t n t dt s t n t dtT T

s t n t j n t dt dtT

s t e dtT

ω

ω ω

ω ω

+ +

+

+−

= − ≥

= −

= −

=

∫ ∫

∫

∫

0

*

( ) leads to

( ) leads to

( ) leads to

v

n

jn tv n

n

k s t k c

s t t c e

s t c

ω

⋅ ⋅

− ⋅

−



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2.3.1 Periodic Signals and the Fourier SeriesThe convergence of the exponential form

0

2

0

lim ( ) 0T

jn tn

n

s t c e dtν

ω

ν ν

+

→∞=−

⎡ ⎤− =⎢ ⎥⎣ ⎦∑∫

1) The Fourier series converges in the mean square average:

2) At finite numbers of jumps in the period T the Fourier seriesapproaches the jump, it is equal to s(t) before and after thejump and crosses the jump at its center

Interpretation of coefficients:

Complex coefficients as a pair represent one signal componentwith a certain frequency of n times the fundamental frequency ω0.

Magnitude and phase of the complex coefficient correspond to amplitude and phase (delay/advance) of that signal component.



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2.3.1 Periodic Signals and the Fourier SeriesThe Gibb‘s Phenomenon

At jumps the Fouries series introduces overshoots into the signal

These overshoots can be observed for low-pass signals e.g.

This effect is always given, even for a perfect Fourier series withinfinetely much components!

Fourier Series with n=11

Periodic Rectangular Series

( )s t

0.5

1

0

0 1 2t



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2.3.1 Periodic Signals and the Fourier SeriesThe distortion factor

( )

( )

0 0

0

t2 22 2

,0 t

2 2

2

2 2

1

1where = ( ) yields inT

T

n eff n n n

n nn

n nn

s c c s t dt P

c cK

c c

+

−

∞

−=

∞

−=

= + =

+=

+

∫

∑

∑

2 2 22, 3, 4,

2 2 2 21, 2, 3, 4,

rms-value of the signal harmonicsrms-value of all harmonics

...

...eff eff eff

eff eff eff eff

K

s s s

s s s s

=

+ + +=

+ + + +

Distortion factor is a measure for amount of higher harmonics in the signal

Note:

DC component is no harmonic



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2.3.2 The Fourier Transform



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2.3.2 The Fourier Transform - Definition

( )s t dt+∞

−∞

< ∞∫

1( ) ( )2

j ts t S e dωω ωπ

+∞

−∞

= ⋅ ⋅∫ ( ) ( ) j tS s t e dtωω+∞

−

−∞

= ⋅∫

0

0 2Tm ω ω

ω π∆

= = ∆

Absolutely integrable signals are denoted by:

For such signals fufilling some additional conditions it holds:

0( ) jn tn

ns t c e ω

+∞

=−∞

= ∑

A periodic signal can be turned into a non-periodic one by extending theperiod to infinite. For periodic signal holds:

In a narrow interval m summation terms (orm lines according to Fourier series) exist:

ω∆



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2.3.2 The Fourier Transform - Definition

( ) ( )ii

s t s t= ∆∑

0 0

2jn t

nTds c e dω ωπ

= ⋅( )s t ds= ∫

0( )F nS T cω = ⋅ 0nω ω=

11( ) ( ) ( )2

j tF Fs t S e d F Sωω ω ω

π

+∞−

−∞

= =∫

The m (nearly not different) lines represent one part of the signal:

0 00( )2

jn t jn ti n n

Ts t m c e c eω ωωπ

∆ ≈ ⋅ = ∆ ⋅

The whole signal then is given by summing up all signal parts:

In the limit (period growing over all limits) the summation turns to the integral:

Now some rewriting is introduced:

Finally the inverse FourierTransform results:



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2.3.2 The Fourier Transform - Interpretation

( ) ( ) ( )j tFS s t e dt F s tωω

+∞−

−∞

= =∫

The Fourier transform is a measure of amplitudes and phases of theharmonics when evaluated at a specific frequency:

( ) and ( )F FS Sω ω

The Fourier Transform is determined by:

This function is also called spectrum or amplitude density spectrum!

The sub F only is used if it is not clear which transform is meant.

Special properties:

All frequencies in a certain interval are present

This transform relates the time-domain and the frequency domain



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2.3.2 The Fourier Transform –Convergence properties

1 ( 0) ( 0)lim ( )2 2

j t s t s tS e dα

ω

αα

ω ωπ→∞

−

+ + −=∫

0 1 ... na t t t b= < < < =10

( ) ( )n

s t s tν νν

−=

− < ∞∑

Convergence properties have to be considered in special cases such as:

- Signals with jumps

- Signals with curves of infinite lenght (no limited variation)

- Signals including Dirac impulses

For absolutely integrable signals with limited variation in suitable intervals it holds:

Limited variation in a finite interval (a,b), which is partitioned means:

(Reasonable point-for-point convergence)

Example: Dirac impulse has no limited variation.



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Additional remarks:

For voltage signals with the dimension of [V] the Fourier transformexhibits the dimension of [V . s] = [ V / Hz ] !

Verify the dimension of the expressions in:

2.3.2 The Fourier Transform- Convergence properties

( 0) ( 0)lim ( )2

j t S Ss t e dtα

ω

αα

ω ω−

→∞−

+ + −⋅ =∫

Fourier transform and inverse transform show up similar relations.

Thus the convergence properties described before can be applied to thefrequency domain:

( ) ( ) j tS s t e dtωω+∞

−

−∞

= ⋅∫



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2.3.2 The Fourier Transform –Another interpretation of the transform values

( ) ( )

0 0

0 0

0 0

0

0

2 2

2 2

0 0

( )

0 0 0 0

0 0

1 1( ) ( ) ( )2 2

( ) ( )2

2Re ( ) 2Re ( )(cos sin )2 2

2Re ( ) cos2

j t j t

j t j t

S

j t

s t S e d S e d

S e S e

S e S t j t

S t

ω ω ω ωω ω

ω ω ω ω

ω ω

ω

ω

ω ω ω ωπ π

ω ω ωπ

ω ωω ω ω ωπ πω ω ωπ

∗

− +∆ +∆

− −∆ −∆

−

∆ = +

⎛ ⎞∆ ⎜ ⎟≈ − +

⎜ ⎟⎜ ⎟⎝ ⎠

∆ ∆= = +

∆=

∫ ∫

( )0 0

0 0 0

2 Im ( ) sin

( ) cos( ( ))

S t

S t S

ω ω

ω ω ω ωπ

−

∆= +

A signal component gained by means of an ideal band pass is considered:

The transform is a measure of amplitude& phase of the signal component!

Smooth form of thespectrum at ω0 is assumed!



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2.3.2 The Fourier Transform – Important properties

1 2( ) ( ) ( )s t s t js t= + ( ) ( ) ( )S R jXω ω ω= +

( )( )1 2( ) ( ) ( ) ( ) cos sinj tS s t e dt s t js t t j t dtωω ω ω+∞ +∞

−

−∞ −∞

= = + −∫ ∫

( )1 2( ) ( ) cos ( )sinR s t t s t t dtω ω ω+∞

−∞

= +∫

( )1 2( ) ( )sin ( ) cosX s t t s t t dtω ω ω+∞

−∞

= − −∫

For complex signals it holds:

Thus it results:



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2 1( ) ( ) cos due to s ( ) 0 and s(t) = s ( )R s t tdt t tω ω+∞

−∞

= =∫

( ) ( )sinX s t tdtω ω+∞

−∞

= − ∫

( ) ( )R Rω ω− = ( ) ( )X Xω ω− = −

( ) ( ) ( ) ( ) ( ) ( )S R jX R jX Sω ω ω ω ω ω∗− = − + − = − =

For real signals some further simplifications can be used:

These integrals show very important properties:

or in other words:

Summary: Real part of the transform is even, imaginary is odd!Magnitude of the transform is even, phase is odd!

Left part of spectrum is conjugated complex compared to right part!



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Note the general mathematical properties of functions:

Any function can be separated

into even and odd parts:


( ) ( ) ( )g us t s t s t= +

( ) ( )( )2g

s t s ts t + −=

( ) ( )( )2u

s t s ts t − −=

( ) ( )g gs t s t− =

( ) ( )u us t s t− = −

with

For these parts it holds:



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2.3.2 The Fourier Transform – Important poperties

( )gs t ( )R ω

( )us t ( )jX ω

0 0

( ) 2 ( ) cos( ) ; ( ) 2 ( ) sin( )g uR s t t dt X s t t dtω ω ω ω+∞ +∞

= ⋅ ⋅ = − ⋅ ⋅∫ ∫

0 0

1 1( ) ( ) cos( ) ; ( ) ( )sin( ) g us t R t dt s t X t dtω ω ω ωπ π

+∞ +∞

= = −∫ ∫

Summary:

If only even or only odd parts of a signal are regarded the Fouriertransform formulas simplify a bit:



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2.3.2 The Fourier Transform – The rules

( )s bt

( )S cω

1 Sb b

ω⎛ ⎞⋅ ⎜ ⎟⎝ ⎠

1 tsc c

⎛ ⎞⋅ ⎜ ⎟⎝ ⎠

For real b,c 0

⎫⎪⎪ ≠⎬⎪⎪⎭

0( )s t t− 0 ( )j te Sω ω− ⋅

0 ( )j te s tω ⋅ 0( )S ω ω−

Rules are important for efficient use of transform tables!

1 Similarity in time- and frequency domain

2 Shifting in the time and frequency domain(delay and modulation)



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( ) ( )ns t ( ) ( )nj Sω ω⋅

( ) ( )nj t s t− ⋅ ⋅ ( ) ( )nS ω

( ) ( )t

g t s dτ τ−∞

= ∫1 ( ) ( )S Gj

ω ωω⋅ =

3 Differentiation in the time and frequency domain

4 Integration in the time domain



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1 2 1 2( ) ( ) ( ) ( )s t s t s s t dτ τ τ+∞

−∞

∗ = ⋅ −∫

1 2( ) ( )s t s t 1 21 ( )* ( )

2S Sω ω

π

1 2( ) ( )s s t dτ τ τ+∞

−∞

⋅ −∫ 1 2( ) ( ) ( )S S Sω ω ω⋅ =

5 Convolution in the time domain / Multiplication in the frequency domain

Abbreviation:

6 Multiplication in the time domain / Convolution in the frequency domain



N T S


1 2 1 21( ) ( ) ( ) ( )

2s t s t dt S S dω ω ω

π

+∞ +∞

−∞ −∞

⋅ = − ⋅∫ ∫

1 2 1 21( ) ( ) ( ) ( )

2s t s t dt S S dω ω ω

π

+∞ +∞∗

−∞ −∞

⋅ = ⋅∫ ∫

1 2( ) ( ) ( )s t s t s t= =

22 1( ) ( )2

s t dt S dω ωπ

+∞ +∞

−∞ −∞

⇒ = ⋅∫ ∫

7 Parseval‘s theorem (for absolutely & squarely integrable signals)

For real signals due to:( ) ( )S Sω ω∗− =

Special case:

Application:

Determination of signalenergy in frequency domain



N T S

2.3.2 The Fourier Transform of special signals

2 ( )sπ ω⋅( )S t−

, sin , cos , ( ) , ( )j te t t t tω ω ω δ ε−

( ) ( )s t a tδ= ⋅+ +

0

- -

S( )= a ( ) ( )j tt e dt a t e dt aωω δ δ∞ ∞

−

∞ ∞

⋅ ⋅ = ⋅ =∫ ∫

0( ) ( )s t a t tδ= ⋅ − 0( ) j tS a e ωω −= ⋅

0j ta e ω⋅ 02 ( )aπ δ ω ω⋅ ⋅ −

Special signals:

Application of shifting in the time domain:

Application of symmetry theorem: 0 0with t ω→



N T S

2.3.2 The Fourier Transform of special signals

0 00( ) cos( ) ( )

2j t j tas t a t e eω ωω −= ⋅ = ⋅ +

[ ]0 0( ) ( ) ( )S aω π δ ω ω δ ω ω= ⋅ ⋅ − + +

0 00( ) sin( ) ( )

2j t j tas t a t e e

jω ωω −= ⋅ = ⋅ −

[ ]0 0( ) ( ) ( )S j aω π δ ω ω δ ω ω= ⋅ + − −

Now a cosine is written by 2 exponential functions. Also this last result is used:

0j ta e ω⋅ 02 ( )aπ δ ω ω⋅ ⋅ −

Same procedure is applied for a sine function:

Thus we obtain:



N T S

2.3.2 The Fourier Transform of special signals( ) ( )0coss t tω=

0

4πω

−0

2πω

−0

4πω0

2πω

0

a

a−

0

aπ

0ω− 0ω

aπ

( ) ( )s Rω ω=

t

ω0

( ) ( )0sins t tω=

0

4πω

−0

2πω

−0

4πω0

2πω0

a

a−

0

jaπ

0ω−0ω

jaπ−

( ) ( )s jXω ω=

t

ω0



N T S

2.3.4 Laplace Transform of Signals

( ) 0 for 0s t t≡ <

1( ) lim ( )2

jpt

Lj

s t S p e dpj

σ ω

ωσ ωπ

+

→∞−

= ⋅ ⋅∫ p jσ ω= +

( ) ( ) where 0 and realts t t e σε σ−⋅ ⋅ >

0

( ) ( ) ptLS p s t e dt

∞−= ⋅∫( )s t

0 0

( ) ( ) ( )pt t j tLS p s t e dt s t e e dtσ ω

∞ ∞− − −= ⋅ = ⋅ ⋅∫ ∫

For causal signals (see following property) the Laplace transform exists.

Interpretation:

Laplace transform is a Fourier transform of the damped causal signal:

Abbreviation similar to Fourier transform:



N T S

Convergence of the Laplace integral

It converges for all s(t) growing slower with t than

If there is convergence in one point p0 , then there is also convergence in all points p with higher real part of p.

The area of convergence is always a half p plane!

Areas with no convergence are of high interest because location of poles isimportant in many aspects!


0

0

2( ) t ts t a rectt

⎛ ⎞−= ⋅ ⎜ ⎟

⎝ ⎠

0( ) (1 )ptL

aS p ep

−= ⋅ −

Example 1:

teσ

( )s t

a

0 0t t



N T S


0( ) ( ).sins t a t tε ω= ⋅

0 02 2

0 0 0

( )( ) ( )L

a aS pp p j p j

ω ωω ω ω

⋅ ⋅= =

+ + ⋅ −

Example 2:

Some first properties of the Laplace transform

The Laplace transform develops to the Fourier transform onthe vertical axis if some conditions are met:

For real p the Laplace transform is also real, if other conditions are met

( ) ( )L FS j Sω ω=



N T S


( )s b t⋅ 1L

pSb b

⎛ ⎞⋅ ⎜ ⎟⎝ ⎠

( )LS c p⋅ 1 tsc c

⎛ ⎞⋅ ⎜ ⎟⎝ ⎠

For real-valued , 0b c

⎫⎪⎪ >⎬⎪⎪⎭

0( )s t t−0 ( )t p

Le S p− ⋅ 0 0t >

0( )s t t+0

0

0

( ) ( )t

t p ptLe S p e s t dt−

⎛ ⎞⋅ − ⋅⎜ ⎟⎜ ⎟⎝ ⎠

∫ 0 0t >

Rules for the Laplace transform

1 Scaling

2 Shifting on the time axis



N T S


0 ( )p te s t− ⋅ 0( )LS p p+

( )d s tdt

( ) (0)Lp S p s⋅ −

( 1) ( )n nt s t− ⋅ ⋅ ( ) ( )nLS p

3 Shifting on the frequency axis

4 Differentiation in the time domain

5 n-times differentiation in the frequency domain



N T S


0

( )t

s dτ τ∫1 ( )LS pp⋅

1( )s t 1( )LS p

2 ( )LS p2 ( )s t

1 20

( ) ( )t

s s t dτ τ τ⋅ −∫ 1 2( ) ( )L LS p S p⋅

6 Integration in the time domain

7 Convolution in the time domain



N T S

2.3.5 Z-Transform of Discrete-TimeSequences

( )s k 0

( ) ( ) ( )kz

kS z s k z Z s k

∞−

=

= ⋅ =∑

0 for 0( )

( ) for k 0k

s ks k

<⎧= ⎨ ≥⎩

( ) ( )as k t s kδ⋅ =

For discrete signals in most cases instead of the Laplace the z-transform is used:

with

This transform results from the Laplace transform for the case of discretesignals with constant clock period.

Here an ideally sampled continous-time (analog) signal sa(t) is assumed.

0

( ) ( ) ( ) s ak

s t s k t t k tδ∞

=

= ⋅∆ ⋅ − ⋅∆∑ All samples of the continous-timesignal can also be written in short:



N T S


0

0

0

( ) ( ) ( )

( ) 1

( )

s ak

pk ta

k

k tpa

k

L s t s k t L t k t

s k t e

s k t e

δ∞

=

∞− ∆

=

∞− ⋅∆

=

= ⋅∆ ⋅ − ⋅∆

= ⋅∆ ⋅ ⋅

= ⋅∆ ⋅

∑

∑

∑

0

( ) ( ) ( )s ak


=

= ⋅∆ ⋅ − ⋅∆∑

3 tt 4 t 5 t 8 t 9 t 10 t

( )ss t

t



N T S


t pz e∆ ⋅= ( ) ( )as k s k t= ⋅∆

0

( ) ( ) ( ) kz

kL s k S z s k z

∞−

=

= =∑

11( ) ( ) 0,1,2,...2

kZ

c

s k S z z dz kjπ

−= =∫

Thus we obtain an expression which is no more directly depending on p:

This is the z-transform. The inverse transform looks as follows:

The exponential expression can also be written in short.



N T S


0

1 for 0( ) ( )

0 for 0k

s k kk

γ=⎧

= = ⎨ ∀ ≠⎩

00

0( ) ( ) 1 1k

zk

S z k z zγ∞

− −

=

= = =∑

1

0 for 0( ) ( )

1 for 0k

s k kk

γ −

<⎧= = ⎨ ≥⎩

10

1( )1 1

kz

k

zS z zz z

∞−

−=

= = =− −∑

Example 1: Unit impulse

Example 2: Unit step sequence



N T S


0 for 0( ) 1 for 0

!

ks k

kk

<⎧⎪= ⎨

≥⎪⎩

1

0( )

!

kz

Zk

zS z ek

−∞

=

= =∑

Example 3:



N T S

2.3.5 Z-Transform of Discrete-Time Sequences

( 1)s k − 1 ( )Zz S z−

( 1)s k + [ ]( ) (0)Zz S z s−

( )akTe s k ( )aTZS e z−

( )k s kα − ZS zα

Rules and properties of the z-transform

1 Shifting

2 Modulation

3 Damping



N T S


( )ks k ( )ZdS zzdz

−

( )s k ( )zS z

( )g k ( )zG z

0

( ) ( ) ( ) ( )k

s k g k s g kν

ν ν=

∗ = −∑ ( ) ( )z zS z G z

4 Differentiation of the z-transform

5 Convolution

6 Linearity

0

( ) ( ) ( ) ( )k

s k g k s g kν

ν ν=

∗ = −∑ ( ) ( )z zS z G z



N T S


1 1( 2) ( 1) ( ) ( 1) ( ) with 2.5y k c y k y k s k s k c− + − + = − + = −

( )y k ( )s k( )ZY Z ( )ZS Z

( ) 0 0s k k= ∀ <( ) 0 0y k k= ∀ <

Example (2nd order processing of an input sequence)

For this situation the output sequence in terms of the input sequenceis required (zero state and causal input sequence is assumed).



N T S


1( 2) ( 1) ( ) ( 1) ( )y k c y k y k s k s k− + − + = − +

2 1 11

2 1 11

1

2 11

( )

( ) ( ) ( ) ( ) ( )

( ) 1 ( ) 1

1 ( ) ( )1

Z

Z Z Z Z Z

Z Z

Z Z

H z

z Y z c z Y z Y z z S z S z

Y z z c z S z z

zY z S zz c z

− − −

− − −

−

− −

+ + = +

⎡ ⎤ ⎡ ⎤⇒ + + = +⎣ ⎦ ⎣ ⎦+

⇒ =+ +

( ) ( ) ( )Z Z ZY z S z H z= ( ) ( ) ( )y k s k h k= ∗

In short the result readsas follows:



N T S


1 12( ) ( 0.5 2 0.5 2 )3

k k k kh k + += − + − +

1 11 or

1

2 2( 1) (0.5 2 ) (0.5 2 )3 3

k k k kk kk k

h k − −− →→ +

− = − − − −

1 1( ) ( 0.5)( 2) ( 0.5)( 2)Z

zH z zz z z z

−⋅ = +− − − −

1 2

2 1 21

1 ( 1)( ) 1 2.5 1 ( 0.5)( 2)

1 ( 0.5)( 2) ( 0.5)( 2)

Zz z z z zH z

z c z z z z z

zzz z z z

−

− −

+ + += = =

+ + − + − −

⎛ ⎞= ⋅ +⎜ ⎟− − − −⎝ ⎠



N T S


( )as t ( )aS ω

due to ( ) ( )as k t s k∆ =

0

( ) ( ) ( )s ak


=

= ∆ − ∆∑

0 0

0

( ) ( ) ( )

Comparison to ( ) ( ) gives the relation:

( ) ( )

j k t j k ts a

k k

kZ

k

j ts Z

S s k t e s k e

S z s k z

S S e

ω ω

ω

ω

ω

∞ ∞− ∆ − ∆

= =

∞−

=

∆

= ∆ =

=

=

∑ ∑

∑

Also for the z-transform the frequency response of a system is of largeimportance. AS before the sampling of an analog signal is considered, but now the Fourier transform is applied:



N T S


j tz e ω∆= 1 for all z tω⇒ = ∆

0( ) ( )j t j t k

Zk

S e s k eω ω∞

∆ − ∆ ⋅

=

= ⋅∑

Conclusion:

To obtain the properties of the discrete signal in the frequency domain the z-transform has to be evaluted only at the following points:

Thus we evaluate the z-transform on the unit-circle:



N T S


0

0

( )0.5a

t ts t Ak t

⎛ ⎞−= ⋅Λ⎜ ⎟⋅ ⋅∆⎝ ⎠

020 0 sinc2 4

j tk k tA t e ωω −⋅∆⎛ ⎞⋅ ⋅∆ ⋅ ⋅ ⋅⎜ ⎟⎝ ⎠

Example (triangular sequence)

( )S k

012

k

A

00k

k

t



N T S

2.4 Important General Signal Representations

( ) ( )R Rω ω= −

( ) ( )X Xω ω= − −

( ) ( )S Sω ω= −

( ) ( )ϕ ω ϕ ω− = −

( ) ( )S Sω ω∗− =

( )s t ( )( ) ( ) ( ) ( ) with ( ) ( )jS R j X S e Sϕ ωω ω ω ω ϕ ω ω= + ⋅ = ⋅ = ∠

If all physical signals in the time domain are real, it follows:

or



N T S

2.4.1 Low-Pass Signals

( ) 0 or ( ) 0 for 2g gS S fωω ω ω π>= ≈ =

„Low-pass signal“ are signals s(t) with a spectrum S(w) that vanishes completelyor negligible for

The spectrum of a low-pass signal exhibits at w=0 always non-zero values



N T S


( )s t

32 gf− 2

2 gf− 1

2 gf− 1

2 gf2

2 gf3

2 gf

t

0gs

ωπ

0

0( )2 g

S S rect ωωω

⎛ ⎞= ⋅ ⎜ ⎟⎜ ⎟

⎝ ⎠0( ) ( )g

gs t S si tω

ωπ

= ⋅ ⋅

Example 1: Ideal low-pass signal

Spectrum S(w) of an Ideal low-pass signal

Ideal low-pass signal s(t)

( )S ω

0S

gω− gω ω



N T S


( ) ( )220b t ts t ae− −=

0 0t

a

t 0 gω

0s

gω− ω

( ) 0tϕ ω ω=−

( )S ω

2 2 2 20 0( ) ( )

0( ) b t t b t tbs t a e S eπ

− ⋅ − − ⋅ −= ⋅ = ⋅ ⋅2 2

2 20(2 ) (2 )

0( ) j tb bS a e e S eb

ω ωωπω

− −−= ⋅ ⋅ ⋅ = ⋅

Example 2: the Gaussian impulse

Time function of the Gaussian Impulse Spectrum S(w) (mag. and phase)



N T S

2.4.2 The Hilbert Transform and the AnalyticSignal

2 ( )s t dt+∞

−∞

< ∞∫

1 ( )ˆ( ) ( ) . . ss t H s t V P dtτ τ

π τ

+∞

−∞

= =−∫

0

( ). lim ... ...t

t

sV P d d dt

ε

εε

τ τ τ ττ

+∞ − +∞

→−∞ −∞ +

⎡ ⎤= +⎢ ⎥− ⎣ ⎦

∫ ∫ ∫

ˆ1 ( ) ˆ( ) . . ( )ss t V P d H s ttτ τ

π τ

+∞

−∞

= − = −−∫

For signals:

the Hilbert transform of the signal s(t) is given as:

where

Accordingly, the inverse Hilbert transform is given by:



N T S


1 ( ) 1ˆ( ) ( )ss t d s tt tτ τ

π τ π

+∞

−∞

= = ∗−∫

ˆ1 ( ) 1ˆ( ) ( )ss t d s tt tτ τ

π τ π

+∞

−∞

= − = − ∗−∫

ˆˆ( ) ( ) ( ) ( )F s t j sign S Sω ω ω= − ⋅ ⋅ = ˆ( ) ( ) ( ) ( )F s t j sign S Sω ω ω= ⋅ ⋅ =

Hilbert transform can be interpreted by means of convolution integrals in case the intergrals converge:

and

Thus, the Fourier transformscan be derived directly:



N T S


If s(t) is real with S( ) ( ) ( ), it gives result:R j Xω ω ω= + ⋅

0 0

1 1( ) ( ) cos( ) ( ) sin( )s t R t d X t dω ω ω ω ω ωπ π

∞ ∞

= ⋅ − ⋅∫ ∫

0 0

1 1ˆ( ) ( ) cos( ) ( ) sin( )s t X t d R t dω ω ω ω ω ωπ π

∞ ∞

= ⋅ + ⋅∫ ∫

ˆ( ) and ( ) are called conjugated functions.s t s t



N T S


0

0

1( ) ( )2

1 1 ( ) ( )2 2

S( ) ( ) ( )

j t

j t j t

s t S e d

S e d S e d

S S

ω

ω ω

ω ωπ

ω ω ω ωπ π

ω ω ω

+∞

−∞

+∞

−∞

− +

= ⋅ ⋅

= ⋅ ⋅ + ⋅ ⋅

= +

∫

∫ ∫

ˆ( ) ( ) ( )s t s t j s t= + ⋅

1 ( )ˆ( ) ss t dt

τ τπ τ

+∞

−∞

= − ⋅−∫

With:

the analytic signal is defined as following:

Real part: the signal itself Imaginary part: Hilbert transform of s(t)



N T S


1. If s(t) S( ), thenω

ˆ( )s t( ) for 0

ˆ( ) 0 for 0( ) for 0

j SS

j S

ω ωω ω

ω ω

− ⋅ >⎧⎪= =⎨⎪+ ⋅ <⎩

ˆ( )s t ˆ( ) ( ) ( )S j S signω ω ω= − ⋅ ⋅

The properties of analytic signal:



N T S


( )s t0 for 0

( ) ( ) for 02 ( ) for 0

S SS

ωω ω ω

ω ω

<⎧⎪= =⎨⎪ >⎩

0

1 1( ) ( ) ( )2

j t j ts t S e d S e dω ωω ω ω ωπ π

+∞ ∞

−∞

= ⋅ = ⋅∫ ∫

ˆ( ) ( ) ( )s t s t js t= +

[ ]

ˆ( ) ( ) ( ) ( ) ( ( ) ( ))0 for 0

( ) 1 ( ) ( ) for 02 ( ) for 0

S S j S S j jsign S

S sign SS

ω ω ω ω ω ωω

ω ω ω ωω ω

= + ⋅ = + −

<⎧⎪= ⋅ + = =⎨⎪ ⋅ >⎩

Proof:

2. With s(t) S( ), result:ω

or



N T S


ˆ( ) ( ) 0s t s t dt+∞

−∞

⋅ =∫

0( ) ( )gg

Ah t si t

ωω

π⋅

= ⋅ ⋅

0 sin 1 cosˆ( ) ( ) ( ) g g g

g g

A t th t h t j h t j

t tω ω ωπ ω ω

⎡ ⎤⋅ −= + ⋅ = ⋅ + ⋅⎢ ⎥

⎢ ⎥⎣ ⎦

Example:

ˆ3. Real part ( ) and imaginary part ( ) are orthogonal:s t s t



N T S


Real and Imaginary part of the analytic Signal of an Ideal low-pass

Example:

0gA

ωπ

( )h t ( )h t

0t



N T S

2.4.3 Band-Pass Signals

( ) 0 or ( ) 0 for all outside of S Sω ω ω ω= ≈ ∆

Band-pass signals are signals s(t) with spectrum S( ) limited to a certain interval on the frequency axis

ω

This interval does not include the frequency w = 0

The two versions of band-pass signal which will be described following are:

• Symmetrical band-pass signal

• More generalized version



N T S


0 00( ) S S rect rectω ω ω ωω

ω ω⎡ − + ⎤⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎢ ⎥∆ ∆⎝ ⎠ ⎝ ⎠⎣ ⎦

0 0

0 0

0

0

0 0

0

( )2 2 2 2

2 2

cos2

( ) cos

j t j t

j t j t

T

s t S si t e si t e

S si t e e

S si t t

s t t

ω ω

ω ω

ω ω ω ωπ πω ωπω ω ωπ

ω

−

−

⎡∆ ∆ ∆ ∆ ⎤⎛ ⎞ ⎛ ⎞= ⋅ ⋅ + ⋅ ⋅⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∆ ∆⎛ ⎞ ⎡ ⎤= ⋅ ⋅ +⎜ ⎟ ⎣ ⎦⎝ ⎠∆ ∆⎛ ⎞= ⋅ ⋅ ⋅⎜ ⎟

⎝ ⎠= ⋅

The symmetric band-pass signal:

( )S ω

ω ω0s

2ω− 0ω− 1ω− 0 2ω1ω 0ω ω

2 fω π=



N T S


( )S ω

ω ω0s

2ω− 0ω− 1ω− 0 2ω1ω 0ω ω

2 fω π=

Example:

Spectrum of a symmetric (Ideal) Band-pass signal

Symmetric (Ideal) Band-pass signal and its envelope

( )s t

t

Envelope

Envelope



N T S


( )s t

t

Envelope

Envelope

4πω

−2πω

− 0 2πω

4πω

0( ) 2TS S rect ωωω

⎛ ⎞= ⋅ ⎜ ⎟∆⎝ ⎠0( ) 2

2 2Ts t S si tω ωπ

∆ ∆⎛ ⎞= ⋅ ⋅ ⎜ ⎟⎝ ⎠

Example: Equivalent low-pass signal and its Spectrum

( )TS ω02S

0Sω ω

0ω− 0ω2ω−

2ω0

ω



N T S


ˆ( ) ( ) ( )s t s t j s t= + ⋅ [ ]( ) ( ) 1 ( ) 2 ( ) ( )S S sign Sω ω ω ω ε ω= ⋅ + = ⋅ ⋅

00

0 for 0( ) ( ) for 0 2

2 ( ) for 0S S S rect

S

ωω ωω ω ω

ωω ω

<⎧−⎪ ⎛ ⎞= = = ⋅ ⋅⎨ ⎜ ⎟∆⎝ ⎠⎪ >⎩

Presentation of symmetrical band-pass signals using equivalent low-pass signals:

Analytic signal of a Symmetricband-pass signal

ω( )S ω°02S

0S

0ω− 0ω0ω



N T S


As ( ) is real, the following relation holds:Ts t

0( ) ( ) j tTs t s t e ω−= ⋅

0 0( ) ( ) 2TS S S rect ωω ω ωω

⎛ ⎞= + = ⋅ ⎜ ⎟∆⎝ ⎠

0( )2T

Ss t si tω ωπ∆ ∆⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠

By shifting on the frequency axis, the equivalent low-pass signal can bederived as:

The equivalent low-pass signal of a Symmetric Band-pass signal

0 00( ) Re ( ) Re ( ) ( ) Re ( ) cosj t j t

T T Ts t s t s t e s t e s t tω ω ω= = ⋅ = = ⋅

( )TS ω

0S

2ω

02S

2ω− 0 ω



N T S


( )TS ω

0( ) ( ) cos( ( )): inphase componentu t s t tφ= ⋅

( )0( ) ( ) ( ) ( ) j t

Ts t u t j v t s t e φ= + ⋅ = ⋅

0( ) ( ) sin( ( )): quadrature componentv t s t tφ= ⋅

The general, real band-pass signal:

Spectrum of a Non-symmetric Real Band-pass signal

Signal envelope

ω

BA

C

C−

B−

0

ω

( )Re S ω( )Im S ω

( )Re S ω

( )Im S ω

( )Im S ω

( )Re S ω

ω



N T S

2.4.3 Band-Pass Signals0One can find the signal ( ) developed from the equation:s t

0by choosing : "midband frequency" as following description in the figure

ω

0Spectrum S ( ) of the Analytic Signal of the Non-symmetric Real band-pass signal

ω

0 0( )( ) ( ) j tTs t s t e ω φ+= ⋅

00( ) ( )j

TS e Sφω ω ω= ⋅ −( )Re S ω°( )Im S ω°

2 A

2C−

2B−

00ω ω

( )Re S ω°

( )Im S ω°

ω

In the follwing we set without loss of generality :

00 0 1je φφ = ⇒ =



N T S


Example:

Complex envelope (Real and Imaginary part) of the Non-symmetric real band-pass signal

( )Re TS ω( )Im TS ω

2A

2C−

2B−

0ω

( )Re TS ω

( )Im TS ω

ω



N T S


1. ( ) ( ) 2 ( )S Sω ω ε ω= ⋅ ⋅

0( ) ( )TS Sω ω ω= +

0( )0 0( ) Re ( ) Re ( ) ( ) cos( ( ))j t

Ts t s t s t e s t t tω ω φ= = ⋅ = ⋅ +

0 0

0 0

0 0 0 0 0

( ) Re ( ) (cos( ) sin( )) ( ) cos( ) ( ) sin( )

For 0 it holds: ( ) ( ) cos( ) ( ) sin( )

Ts t s t t j tu t t v t t

s t u t t v t t

ω ωω ω

φ ω φ ω φ

= ⋅ + ⋅

= ⋅ − ⋅

≠ = ⋅ + − ⋅ +

In general some followings relations hold:

2. The spectrum‘s relationship between complex envelope and analyticsignal is as follows:

3. The band-pass signal s(t) can be represented in the form:

or



N T S

2.4.4 Causal Signal Functions

( ) 0 for 0s t t≡ <

( ) 0 for k 0s k ≡ <

A causal signal function has the property:

for analog signal

for discrete signal

Causal, analog signals s(t):



N T S


( ) ( )ats t e tε−= 1( ) is causal for 0LS p ap a

= >+

For ( )s t 1 2( ) ( ) ( ), applies:S S jSω ω ω= +

2 1( ) ( )S Sω ω= 1 2ˆ( ) ( )S Sω ω= −

Example:

The unique relation between real part and imaginary part of causal signal spectra:

and

1 1( ) ( ) ( )2

1 1 1( ) ( ) ( )2 21 1 1( ) ( )

2 2

S Sj

S Sj

S Sj

ω ω πδ ωπ ω

ω ω δ ωπ ω

ω ωπ ω

⎡ ⎤⎛ ⎞= ∗ +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

= ∗ + ∗

= ∗ +



N T S


!( ) ( ) ( )s t s t tε=

1 1( ) ( ) ( )2

1 1 1( ) ( ) ( )2 21 1 1( ) ( )

2 2

S Sj

S Sj

S Sj

ω ω πδ ωπ ω

ω ω δ ωπ ω

ω ωπ ω

⎡ ⎤⎛ ⎞= ∗ +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

= ∗ + ∗

= ∗ +



N T S


1 2( ) ( ) ( )S S jSω ω ω= +

1 2 1 21 1 1( ) ( ) ( ) ( )S jS S jSj

ω ω ω ωπ ω ω⎡ ⎤+ = ∗ + ∗⎢ ⎥⎣ ⎦

1 1( ) ( )S Sj

ω ωπ ω

= ∗



N T S


12 1 2

( )1 1 1 ˆ( ) ( ) ( )SS S d Sηω ω η ωπ ω π ω η

+∞

−∞

= − ∗ = − = −−∫

21 2 1

( )1 1 1 ˆ( ) ( ) ( )SS S d Sηω ω η ωπ ω π ω η

+∞

−∞

= ∗ = =−∫

0

( ) where 1 and arbitrary realks k M M∞

− < ∞ >∑



N T S


000 0 0

for 0( ) where and 1

0 for 0

kjz k

s k z z e zk

φ⎧ ≥= = ≤⎨

<⎩

0

( )ZzS z

z z=

−

Re Z

Im Z

0Z

1−

1−

1+

1+

Z - plane



N T S


jz e Ω= 1z =

( )s k ( ) ( ) ( ) ( ) ( ) ( )j j jZ Z Z Z Z ZS z R z jX z S e R e jX eΩ Ω Ω= + ⇒ = +

1

1( ) (0) ( )sin( ) cos( )j jZ Z

k

R e s X e k d kπ

η

π

η ηπ

+∞Ω

= −

⎡ ⎤= − Ω⎢ ⎥

⎣ ⎦∑ ∫

1

1( ) ( ) cos( ) sin( )j jZ Z

k

X e R e k d kπ

η

π

η ηπ

+∞Ω

= −

⎡ ⎤= − Ω⎢ ⎥

⎣ ⎦∑ ∫



N T S

2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals

2( )sE s t dt+∞

−∞

= < ∞∫( )( )n

s

s ts tE

=( )( )n

g

g tg tE

=

2( ( ) ( )) 2s g n n sgE s t g t dt r+∞

−−∞

= − = −∫

( ) ( )

sgs g

s t g t dtr

E E

+∞

−∞=∫



N T S


Re Im( ) ( ) ( )s t s t js t= +

2( ) ( ) ( )sE s t s t dt s t dt+∞ +∞

∗

−∞ −∞

= =∫ ∫

( ) ( )

sgs g

s t g t dtr

E E

+∞∗

−∞=∫

1 1sgr− ≤ ≤ +

( ) ( )s t kg t= 2s gE k E= 1sgr = +



N T S


( ) ( )s t kg t= −

2s gE k E= 1sgr = −

0sgr =

( ) ( ) ( )sg s t g t dtρ τ τ+∞

∗

−∞

= +∫

( ) ( ) ( ) ( ) ( )sg s t g t dt s gρ τ τ τ τ+∞

∗

−∞

= + = ⊗∫



N T S


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sg s g d s g d s gρ τ θ θ τ θ θ τ θ θ τ τ−∞ +∞

∗ ∗

+∞ −∞

= − − + − = − − = − ∗∫ ∫

( )sgρ τ ( ) ( ) ( ) ( )jsg sgR e d S Gωτω ρ τ τ ω ω

+∞− ∗

−∞

= = − −∫( ) ( ) ( )sg s t g tρ τ ∗= − ∗

( ) ( ) ( )sgR S Gω ω ω∗= − ∗ −

( )f t− ( )F ω−

( )f t∗ ( )F ω∗ −



N T S


( ) ( )sg gsρ τ ρ τ∗− = ( ) ( )sg gsR Rω ω∗− = −

( ) ( )sg gsρ τ ρ τ∗= − ( ) ( )sg gsR Rω ω∗=

( ) ( ) ( ) ( )s t g t g t s t⊗ ≠ ⊗

( ) ( ) ( ) ( )s t g t g t s t⊗ = − ⊗ −



N T S


( ) ( ) ( ) ( ) ( )ss s t s t dt s t s tρ τ τ+∞

∗

−∞

= + = ⊗∫

2( ) ( ) ( ) ( )ssR S S Sω ω ω ω∗= − = −

( ) ( ) ( ) ( ) ( )ss s t s t s t s tρ τ = ⊗ = − ∗

( ) ( ) ( ) ( ) ( )ss t s t s t s t s tρ = ⊗ = − ∗

2( ) ( ) ( ) ( )ssR S S Sω ω ω ω∗= =



N T S


2 2

(0) ( ) ( ) (signal energy of s(t))

1 1 ( ) ( ) (2 )02 2

ss s

j tss

s t s t dt E

R e d S d S f dft

ω

ρ

ω ω ω ω ππ π

+∞

−∞

+∞ +∞ +∞

−∞ −∞ −∞

= =

= = ==

∫

∫ ∫ ∫



N T S


( )s t

0t0t− 0t

1

( )02

ts t rectt

⎛ ⎞= ⎜ ⎟

⎝ ⎠

( )g t

0t0t− 0t

1

( )0 0

sin2 2t tg t rectt t

π⎛ ⎞ ⎛ ⎞

= ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

1−

02t02t−

0

0 0

( ) ( ) ( ) ( ) ( )

2 sin2 4

sg s t g t s t g t dt

t rectt t

ρ τ τ

πτ τπ

+∞∗

−∞

= ⊗ = +

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∫

0

0 0

( ) ( ) ( )

2 sin2 4

gs sg sg

t rectt t

ρ τ ρ τ ρ τ

πτ τπ

∗= − = −

⎛ ⎞ ⎛ ⎞= − ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Two deterministic energy signal s(t) and g(t)



N T S


02t

02t−

02t02t−

02tπ

02tπ

−

02tπ

02tπ

−

( )sgρ τ

( )gsρ τ

τ

τ

Example 1: (cont.)

The resulted Cross-correlation function for s(t) and g(t)



N T S


0

0 0

( ) ( ) ( ) ( ) ( ) 2 sin2 4sg

ts t g t s t g t dt rectt tπτ τρ τ τ

π

+∞∗

−∞

⎛ ⎞ ⎛ ⎞= ⊗ = + = ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∫

0

0 0

( ) ( ) ( ) 2 sin2 4gs sg sg

t rectt tπτ τρ τ ρ τ ρ τ

π∗ ⎛ ⎞ ⎛ ⎞

= − = − = − ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



N T S


02t− 02t0t

0t0t−

0t−

0

0 τ

τ

( )0

12

s t A rectt

⎛ ⎞= ⎜ ⎟

⎝ ⎠ ( )s t

( )ssρ τ2

02A t⋅



N T S


0 0

20

0

( ) ( ) ( ) ( ) ( ) . .2 2

24

sst ts t s t s t s t A rect A rectt t

A tt

ρ τ

τ

∗ ⎛ ⎞ ⎛ ⎞= ⊗ = − ∗ = − ∗⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞

= Λ⎜ ⎟⎝ ⎠

2 2 20 0( ) 4 ( )ssR A t si tω ω=



N T S

2.5.2 Cross Correlation Function, Autocorrelation Function and Power spectrum

of Deterministic Analog Power Signals21lim ( )

2

T

s TT

P s t dtT

+

→∞−

= < ∞∫

1( ) lim ( ) ( ) ( ) ( )2

T

sg TT

s t g t dt s t g tT

ρ τ τ+

∗

→∞−

= + = ⊗∫

( ) ( ) jsg sgR e dωτω ρ τ τ

+∞−

−∞

= ∫1( ) ( )

2j t

sg sgR e dωρ τ ω ωπ

+∞

−∞

= ∫



N T S

2.5.2 Cross Correlation Function, Autocorrelation Function and Power spectrum

of Deterministic Analog Power Signals

( ) ( )T

j tT

T

S s t e dtωω+

−

−

= ∫ ( ) ( )T

j tT

T

G g t e dtωω+

−

−

= ∫

( )1( ) lim ( ) ( )2sg T

s t g tT

ρ τ ∗

→∞= − ∗ 1( ) lim ( ) ( )

2sg T TTR S G

Tω ω ω∗

→∞= − −

( ) ( )sg gsρ τ ρ τ= −

( ) ( )sg gsR Rω ω∗=

Signals and Systems 1 -...

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