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Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 1
FachgebietNachrichtentechnische Systeme
N T S
Signals and Systems 1
Prof. Dr.-Ing. I.Willms
(Version 2.1)
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 2
FachgebietNachrichtentechnische Systeme
N T S
Contents
1 Introduction
2 Signal representations in the time- and frequency
domain
3 Analog systems
4 Discrete systems
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 3
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 4
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 5
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 6
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 7
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 8
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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
+ ⋅
⋅
= = = ⋅
= ⋅ + ⋅ ⋅
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 9
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2.2.3 The Dirac Function
0
1( ) limT
trectT T
δ τ→
⎛ ⎞= ⎜ ⎟⎝ ⎠ 0
( )δ τ
τ
Approximation:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 10
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 11
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 12
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 13
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 14
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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+∞
=−∞
= −∑
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 15
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 16
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 17
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2.2.8 Impulse Type Signals
2( / )( ) t Ts t e−=( )s t
3− 2− 1− 32100
1
/t T
The Gaussian impulse
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 18
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2.2.8 Impulse Type Signals
1 for 0( ) ( ) 0 for 0
1 for 0
ts t sign t t
t
>⎧⎪= = =⎨⎪− <⎩
( )sign t
t
1
1−
0
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 19
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2.2.8 Impulse Type Signals
1 for ( )
0 otherwise
t t Tts t TT
⎧− ≤⎪⎛ ⎞= Λ = ⎨⎜ ⎟
⎝ ⎠ ⎪⎩
( ) ts tT
⎛ ⎞= Λ⎜ ⎟⎝ ⎠
TT−t
1
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 20
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 21
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 22
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2.2.9 Adjustment of Time and FrequencyFunctions
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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 23
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2.2.9 Adjustment of Time and FrequencyFunctions
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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 24
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2.2.9 Adjustment of Time and FrequencyFunctions
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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 25
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2.2.9 Adjustment of Time and FrequencyFunctions
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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 26
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2.2.9 Adjustment of Time and FrequencyFunctions
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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 27
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2.2.9 Adjustment of Time and FrequencyFunctions
( )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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 28
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2.2.9 Adjustment of Time and FrequencyFunctions
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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 29
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2.2.9 Adjustment of Time and FrequencyFunctions
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.
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 30
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 31
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 32
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 33
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2.3.1 Periodic Signals and the Fourier Series
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!
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 34
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2.3.1 Periodic Signals and the Fourier Series
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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 35
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 36
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2.3.1 Periodic Signals and the Fourier SeriesThe exponential form
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)
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 37
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Additional Fourier series properties:
Linearity
Time delay
Reversal
2.3.1 Periodic Signals and the Fourier SeriesThe exponential form
( )
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
⋅ ⋅
− ⋅
−
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 38
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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.
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 39
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 41
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2.3.2 The Fourier Transform
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 42
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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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2.3.2 The Fourier Transform – Important properties
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:
2.3.2 The Fourier Transform – Important properties
( ) ( ) ( )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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2.3.2 The Fourier Transform – The rules
( ) ( )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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2.3.2 The Fourier Transform – The rules
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
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2.3.2 The Fourier Transform – The rules
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
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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→
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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:
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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
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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:
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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!
2.3.4 Laplace Transform of Signals
0
0
2( ) t ts t a rectt
⎛ ⎞−= ⋅ ⎜ ⎟
⎝ ⎠
0( ) (1 )ptL
aS p ep
−= ⋅ −
Example 1:
teσ
( )s t
a
0 0t t
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2.3.4 Laplace Transform of Signals
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=
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2.3.4 Laplace Transform of Signals
( )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
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2.3.4 Laplace Transform of Signals
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
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2.3.4 Laplace Transform of Signals
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
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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:
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2.3.5 Z-Transform of Discrete-TimeSequences
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
s t s k t t k tδ∞
=
= ⋅∆ ⋅ − ⋅∆∑
3 tt 4 t 5 t 8 t 9 t 10 t
( )ss t
t
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2.3.5 Z-Transform of Discrete-TimeSequences
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.
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2.3.5 Z-Transform of Discrete-TimeSequences
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
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2.3.5 Z-Transform of Discrete-TimeSequences
0 for 0( ) 1 for 0
!
ks k
kk
<⎧⎪= ⎨
≥⎪⎩
1
0( )
!
kz
Zk
zS z ek
−∞
=
= =∑
Example 3:
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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
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2.3.5 Z-Transform of Discrete-TimeSequences
( )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
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2.3.5 Z-Transform of Discrete-TimeSequences
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).
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2.3.5 Z-Transform of Discrete-TimeSequences
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:
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2.3.5 Z-Transform of Discrete-TimeSequences
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
−
− −
+ + += = =
+ + − + − −
⎛ ⎞= ⋅ +⎜ ⎟− − − −⎝ ⎠
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2.3.5 Z-Transform of Discrete-TimeSequences
( )as t ( )aS ω
due to ( ) ( )as k t s k∆ =
0
( ) ( ) ( )s ak
s t s k t t k tδ∞
=
= ∆ − ∆∑
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:
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2.3.5 Z-Transform of Discrete-TimeSequences
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:
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2.3.5 Z-Transform of Discrete-TimeSequences
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
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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
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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
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2.4.1 Low-Pass Signals
( )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
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2.4.1 Low-Pass Signals
( ) ( )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)
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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:
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2.4.2 The Hilbert Transform and the AnalyticSignal
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:
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2.4.2 The Hilbert Transform and the AnalyticSignal
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
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2.4.2 The Hilbert Transform and the AnalyticSignal
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)
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2.4.2 The Hilbert Transform and the AnalyticSignal
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:
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2.4.2 The Hilbert Transform and the AnalyticSignal
( )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
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2.4.2 The Hilbert Transform and the AnalyticSignal
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
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2.4.2 The Hilbert Transform and the AnalyticSignal
Real and Imaginary part of the analytic Signal of an Ideal low-pass
Example:
0gA
ωπ
( )h t ( )h t
0t
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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
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2.4.3 Band-Pass Signals
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=
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2.4.3 Band-Pass Signals
( )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
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2.4.3 Band-Pass Signals
( )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
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2.4.3 Band-Pass Signals
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
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2.4.3 Band-Pass Signals
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
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2.4.3 Band-Pass Signals
( )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
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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 = ⇒ =
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2.4.3 Band-Pass Signals
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
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2.4.3 Band-Pass Signals
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
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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):
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2.4.4 Causal Signal Functions
( ) ( )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
⎡ ⎤⎛ ⎞= ∗ +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
= ∗ + ∗
= ∗ +
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2.4.4 Causal Signal Functions
!( ) ( ) ( )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
⎡ ⎤⎛ ⎞= ∗ +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
= ∗ + ∗
= ∗ +
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2.4.4 Causal Signal Functions
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
= ∗
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2.4.4 Causal Signal Functions
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∞
− < ∞ >∑
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2.4.4 Causal Signal Functions
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
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2.4.4 Causal Signal Functions
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⎢ ⎥
⎣ ⎦∑ ∫
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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
+∞
−∞=∫
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
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 = +
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
( ) ( )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+∞
∗
−∞
= + = ⊗∫
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )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∗ −
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
( ) ( )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⊗ = − ⊗ −
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
( ) ( ) ( ) ( ) ( )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∗= =
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
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
+∞
−∞
+∞ +∞ +∞
−∞ −∞ −∞
= =
= = ==
∫
∫ ∫ ∫
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
( )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)
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
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)
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2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
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∗ ⎛ ⎞ ⎛ ⎞
= − = − = − ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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FachgebietNachrichtentechnische Systeme
N T S
2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
02t− 02t0t
0t0t−
0t−
0
0 τ
τ
( )0
12
s t A rectt
⎛ ⎞= ⎜ ⎟
⎝ ⎠ ( )s t
( )ssρ τ2
02A t⋅
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 118
FachgebietNachrichtentechnische Systeme
N T S
2.5.1 Correlation Functions and Energy Spec. of Deterministic Analog Egergy Signals
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=
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 119
FachgebietNachrichtentechnische Systeme
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
+∞
−∞
= ∫
Prof. Dr.-Ing. I. Willms Signals and Systems 1 S. 120
FachgebietNachrichtentechnische Systeme
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∗=