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LTI Systems, Probability
Analog and Digital Communications
Autumn 2005-2006
Sep 16, 2005
CS477: Analog and Digital Communications
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Sampling

First consider modulation
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Next consider sampling
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Product with Cosine in time domain
Convolution with two impulses in frequency domain
Product with a train of impulses in time domain
Convolution with a train of impulses in the frequency domain
Nyquist sampling theorem

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A bandlimited signal [-B, +B] can be characterized by its
samples taken every 1/(2B) seconds. i.e., 2B samples per
second
Undersampling leads to aliasing
Sep 16, 2005
CS477: Analog and Digital Communications
2
LTI Systems

Linearity
If
then
x (t )
x (t ) =
y( t ) =
P
Pn
cnx n( t )
cnG[x n( t )]
G
y( t )
y( t ) = G[x ( t )]
n

Time invariance
G[x ( t à ü)] = y( t à ü)

Linearity and Time invariance
y( t à ü) =
P
cnG[x n( t à ü)]
n
Sep 16, 2005
CS477: Analog and Digital Communications
3
Response of LTI Systems
Impulse response: h( t ) = G[î ( t )]
y( t ) = G[x ( t )] = G[x ( t ) ã î ( t )]
h R
i
1
= G
x ( ü) î ( t à ü) dü
=
=
1
R
à1
1
R
à1
x ( ü) G[î ( t à ü)]dü
x ( ü) h( t à ü) dü
à1
= x ( t ) ã h( t )
Sep 16, 2005
CS477: Analog and Digital Communications
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Exponentials and LTI Systems
x ( t ) = ej 2ùf ct
Let
y( t ) = x ( t ) ã h( t )
1
R
=
h( ü) x ( t à ü) dü
=
à1
1
R
h( ü) ej 2ùf c(t à ü) dü
Exponentials are
eigenfunctions of
LTI systems!
à1
j 2ùf ct
= e
1
R
h( ü) eà j 2ùf cüdü
à1
= ej 2ùf ct H ( f c)
Sep 16, 2005
LTI Systems can
not generate new
frequencies!
CS477: Analog and Digital Communications
5
Hilbert Transformer


A filter introducing a constant delay of 90 degrees
to the input signal
Hilbert transform does not change the domain;
It’s merely a convolution
ú
à j
f> 0
H Q( f ) = à j sgn( f ) =
+ j
f< 0
x Q( t ) =
1
ùt
xê( t ) = x ( t ) ã hQ( t ) = x ( t ) ã ùt1
xê( t ) $ à j sgn( f ) X ( f )
Sep 16, 2005
CS477: Analog and Digital Communications
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Random Variables


Outcomes and sample space
Random Variables

Mapping outcomes to:
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Discrete numbers  Discrete RVs
Real line  Continuous RVs
Cumulative distribution function
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One variable
Joint cdf
Sep 16, 2005
CS477: Analog and Digital Communications
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Random Variables
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Probability mass function (discrete RV)
Probability density function (cont. RV)
Joint pdf of independent RVs
Mean
Variance
Characteristic function

j 2ùf X
Ð( X ) = E [e
Sep 16, 2005
]=
R
j 2ùf x
e
dx (IFT of pdf)
x
CS477: Analog and Digital Communications
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