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TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Lecture 2:Typicality
Copyright G. Caire (Sample Lectures) 55
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Typical Sequences
• Let x 2 X n. The empirical pmf of x is defined as
⇡(x|x) =
|{i : xi = x}|n
, for x 2 X
This is also referred to as the “type” of x.
• Let Xn denote an i.i.d. random vector with Xi ⇠ PX. By the (weak) law oflarge numbers
lim
n!1⇡(x|Xn
)
p
= PX(x), for x 2 X
Definition 7. Typical set: For a given pmf PX on X and ✏ > 0, the ✏-typical setof sequences x 2 X n is defined as
T (n)
✏ (X) = {x 2 X n: |⇡(x|x) � PX(x)| ✏PX(x), 8 x 2 X}
⌃
Copyright G. Caire (Sample Lectures) 56
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Elementary Properties of Typical Sequences
Lemma 6. Typical Average Lemma: Let x 2 T (n)
✏ (X) and let g(·) denote afunction on X for which E[g(X)] is well-defined (i.e.,
P
x2X PX(x)|g(x)| 1).Then,
(1 � ✏)E[g(X)] 1
n
nX
i=1
g(xi) (1 + ✏)E[g(X)]
⇤
(Proof: HOMEWORK.)
Lemma 7. Asymptotic Equipartition Property (AEP): All typical sequenceshave roughly the same probability. For each x 2 T (n)
✏ (X) we have:
2
�n(H(X)+�(✏)) PXn(x) 2
�n(H(X)��(✏))
where �(✏) # 0 as ✏ ! 0. In short, we write PXn(x)
.= 2
�nH(X). ⇤
(Proof: HOMEWORK.).
Copyright G. Caire (Sample Lectures) 57
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Properties of the Typical Set
• Typical set cardinality upper bound:�
�
�
T (n)
✏ (X)
�
�
�
2
n(H(X)+�(✏))
(Proof: HOMEWORK.)
• Law of Large Numbers (LLN): if Xn is an i.i.d. sequence with Xi ⇠ PX(x)
thenlim
n!1P
⇣
Xn 2 T (n)
✏ (X)
⌘
= 1
• Typical set cardinality lower bound:�
�
�
T (n)
✏ (X)
�
�
�
� (1 � ✏)2n(H(X)��(✏))
for sufficiently large n.(Proof: HOMEWORK.)
Copyright G. Caire (Sample Lectures) 58
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Intuitive Representation
• Typical Average Lemma: Let xn 2 T (n)
✏ (X). Then for any nonnegative functiong(x) on X ,
(1 � ✏) E(g(X)) 1
n
nX
i=1
g(xi) (1 + ✏) E(g(X))
Proof: From the definition of the typical set,�
�
�
�
�
1
n
nX
i=1
g(xi) � E(g(x))
�
�
�
�
�
=
�
�
�
�
�
X
x
⇡(x|xn)g(x) �
X
x
p(x)g(x)
�
�
�
�
�
X
x
✏ p(x)g(x)
= ✏ · E(g(X))
• Properties of typical sequences:
1. Let p(xn) =
Qni=1
pX(xi). Then, for each xn 2 T (n)
✏ (X)
2
�n(H(X)+�(✏)) p(xn) 2
�n(H(X)��(✏)),
where �(✏) = ✏ · H(X) ! 0 as ✏ ! 0. This follows from the typical averagelemma by taking g(x) = � log pX(x)
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 15
2. The cardinality of the typical set�
�T (n)
✏
�
� 2
n(H(X)+�(✏)). This can be shownby summing the lower bound in the previous property over the typical set
3. If X1
,X2
, . . . are i.i.d. with Xi ⇠ pX(xi), then by the LLN
P�
Xn 2 T (n)
✏
! 1
4. The cardinality of the typical set�
�T (n)
✏
�
� � (1 � ✏)2n(H(X)��(✏)) for nsu�ciently large. This follows by property 3 and the upper bound in property1
• The above properties are illustrated in the following figure
X n
T (n)
✏ (X)
p(xn)
.= 2
�nH(X)
|T (n)
✏ | .= 2
nH(X)
P(T (n)
✏ ) � 1 � ✏
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 16
Copyright G. Caire (Sample Lectures) 59
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Jointly Typical Sequences
• Let x,y 2 X n ⇥ Yn. The empirical joint pmf of (x,y) is defined as
⇡(x, y|x,y) =
|{i : (xi, yi) = (x, y)}|n
, for (x, y) 2 X ⇥ Y
Definition 8. Jointly typical set: For a joint pmf PX,Y (x, y) and ✏ > 0, the jointly✏-typical set of sequence pairs (x,y) 2 X n ⇥ Yn is defined as
T (n)
✏ (X, Y ) = {(x,y) 2 X n ⇥ Yn: |⇡(x, y|x,y) � PX,Y (x, y)| ✏PX,Y (x, y),
8 (x, y) 2 X ⇥ Y}
⌃
Copyright G. Caire (Sample Lectures) 60
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Properties of the Jointly Typical Set
• Let (Xn, Y n) be a jointly distributed, componentwise i.i.d., pair of random
vectors with (Xi, Yi) ⇠ PX,Y (x, y), and let (x,y) 2 T (n)
✏ (X, Y ), then thefollowing properties hold:
1. x 2 T (n)
✏ (X) and y 2 T (n)
✏ (Y ).2. PXn,Y n
(x,y)
.= 2
�nH(X,Y ).3. PXn
(x)
.= 2
�nH(X) and PY n(y)
.= 2
�nH(Y ).4. PXn|Y n
(x|y)
.= 2
�nH(X|Y ) and PY n|Xn(y|x)
.= 2
�nH(Y |X).
(Proof: HOMEWORK.)
Copyright G. Caire (Sample Lectures) 61
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Size of Conditional Typical Set
• Let
T (n)
✏ (Y |x) =
n
y 2 Yn: (x,y) 2 T (n)
✏ (X, Y )
o
Then
�
�
�
T (n)
✏ (Y |x)
�
�
�
2
n(H(Y |X)+�(✏))
(Proof: HOMEWORK.)
Copyright G. Caire (Sample Lectures) 62
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Conditional Typicality
Lemma 8. Conditional typicality lemma: Let ✏ > ✏0 > 0. For x 2 T (n)
✏0 (X), letY n ⇠ PY n|Xn
(y|x) =
Qni=1
PY |X(yi|xi). Then
lim
n!1P
⇣
(x, Y n) 2 T (n)
✏ (X, Y )
�
�
�
Xn= x
⌘
= 1
Proof:
In the proof below, the probability measure P(·) denotes the measure of Y n
conditioned on Xn= x (conditioning is omitted for brevity). This is because by
assumption we have Y n ⇠ PY n|Xn(y|x) =
Qni=1
PY |X(yi|xi). The statement isproved if we show that
lim
n!1P (|⇡(x, y|x, Y n
) � PX,Y (x, y)| > ✏PX,Y (x, y) for some (x, y)) = 0
Copyright G. Caire (Sample Lectures) 63
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
For x 2 X such that PX(x) > 0, consider
P (|⇡(x, y|x, Y n) � PX,Y (x, y)| > ✏PX,Y (x, y)) =
= P✓
�
�
�
�
⇡(x, y|x, Y n)
PX(x)
� PY |X(y|x)
�
�
�
�
> ✏PY |X(y|x)
◆
= P✓
�
�
�
�
⇡(x, y|x, Y n)⇡(x|x)
PX(x)⇡(x|x)
� PY |X(y|x)
�
�
�
�
> ✏PY |X(y|x)
◆
= P✓
�
�
�
�
⇡(x, y|x, Y n)
⇡(x|x)PY |X(y|x)
· ⇡(x|x)
PX(x)
� 1
�
�
�
�
> ✏
◆
P✓
⇡(x, y|x, Y n)
⇡(x|x)PY |X(y|x)
· ⇡(x|x)
PX(x)
> 1 + ✏
◆
+ P✓
⇡(x, y|x, Y n)
⇡(x|x)PY |X(y|x)
· ⇡(x|x)
PX(x)
< 1 � ✏
◆
Copyright G. Caire (Sample Lectures) 64
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Since x 2 T (n)
✏0 (X), then
1 � ✏0 ⇡(x|x)
PX(x)
1 + ✏0
Hence
P✓
⇡(x, y|x, Y n)
⇡(x|x)PY |X(y|x)
· ⇡(x|x)
PX(x)
> 1 + ✏
◆
P✓
⇡(x, y|x, Y n)
⇡(x|x)
>1 + ✏
1 + ✏0PY |X(y|x)
◆
Similarly
P✓
⇡(x, y|x, Y n)
⇡(x|x)PY |X(y|x)
· ⇡(x|x)
PX(x)
< 1 � ✏
◆
P✓
⇡(x, y|x, Y n)
⇡(x|x)
<1 � ✏
1 � ✏0PY |X(y|x)
◆
Since by assumption we have ✏0 < ✏, then
1 � ✏
1 � ✏0 < 1 <1 + ✏
1 + ✏0
Copyright G. Caire (Sample Lectures) 65
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Furthermore, since Y n ⇠ Qni=1
PY |X(yi|xi), by the law of large numbers wehave
⇡(x, y|x, Y n)
⇡(x|x)
p! PY |X(y|x)
Hence, both the above upper bounds tend to 0 as n ! 1. Taking the unionover all (x, y) 2 X ⇥ Y and using the union bound yields the final result.
Copyright G. Caire (Sample Lectures) 66
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Consequences of Conditional Typicality
Corollary 7. Let ✏ > ✏0 > 0. For all x 2 T (n)
✏0 (X) and sufficiently large n, wehave �
�
�
T (n)
✏ (Y |x)
�
�
�
� (1 � ✏)2n(H(Y |X)��(✏))
(Proof: HOMEWORK.) Hint: notice that, by definition,
P⇣
(x, Y n) 2 T (n)
✏ (X, Y )
�
�
�
Xn= x
⌘
= P⇣
Y n 2 T (n)
✏ (Y |x)
�
�
�
Xn= x
⌘
and notice that the Conditional Typicality Lemma implies that, for sufficientlylarge n,
P⇣
(x, Y n) 2 T (n)
✏ (X, Y )
�
�
�
Xn= x
⌘
� 1 � ✏
Copyright G. Caire (Sample Lectures) 67
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Intuitive Visualization of Joint Typicality
• Conditional Typicality Lemma: Let xn 2 T (n)
✏0 (X) and Y n ⇠ Qni=1
pY |X(yi|xi).Then for every ✏ > ✏0,
P{(xn, Y n) 2 T (n)
✏ (X,Y )} ! 1 as n ! 1This follows by the LLN. Note that the condition ✏ > ✏0 is crucial to apply theLLN (why?)
The conditional typicality lemma implies that for all xn 2 T (n)
✏0 (X)
|T (n)
✏ (Y |xn)| � (1 � ✏)2n(H(Y |X)��(✏)) for n su�ciently large
• In fact, a stronger statement holds: For every xn 2 T (n)
✏ (X) and n su�cientlylarge,
|T (n)
✏ (Y |xn)| � 2
n(H(Y |X)��0(✏)),
for some �0(✏) ! 0 as ✏ ! 0
This can be proved by counting jointly typical yn sequences (the method oftypes [12]) as shown in the Appendix
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 19
Useful Picture
xn
yn
T (n)
✏ (Y )�| · | .
= 2
nH(Y )
�
T (n)
✏ (X)
�| · | .
= 2
nH(X)
�
T (n)
✏ (X, Y )�| · | .
= 2
nH(X,Y )
�
T (n)
✏ (Y |xn)�
| · | .= 2
nH(Y |X)
� T (n)
✏ (X|yn)�
| · | .= 2
nH(X|Y )
�
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 20Copyright G. Caire (Sample Lectures) 68
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
Intuitive Visualization of Typical Fan-Out
Another Useful Picture
T (n)
✏ (X)
xn
X n Yn T (n)
✏ (Y )
T (n)
✏ (Y |xn)
| · | .= 2
nH(Y |X)
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 21
Joint Typicality for Random Triples
• Let (X,Y, Z) ⇠ p(x, y, z). The set T (n)
✏ (X1
,X2
, X3
) of ✏-typical n-sequencesis defined by
{(xn, yn, zn) :|⇡(x, y, z |xn, yn, zn
) � p(x, y, z)| ✏ · p(x, y, z)
for all (x, y, z) 2 X ⇥ Y ⇥ Z}• Since this is equivalent to the typical set of a single “large” random variable
(X,Y, Z) or a pair of random variables ((X,Y ), Z), the properties of jointtypical sequences continue to hold
• For example, if p(xn, yn, zn) =
Qni=1
pX,Y,Z(xi, yi, zi) and
(xn, yn, zn) 2 T (n)
✏ (X,Y, Z), then
1. xn 2 T (n)
✏ (X) and (yn, zn) 2 T (n)
✏ (Y, Z)
2. p(xn, yn, zn)
.= 2
�nH(X,Y,Z)
3. p(xn, yn|zn)
.= 2
�nH(X,Y |Z)
4. |T (n)
✏ (X|yn, zn)| .
= 2
nH(X|Y,Z) for n su�ciently large
LNIT: Information Measures and Typical Sequences (2010-06-22 08:45) Page 2 – 22
Copyright G. Caire (Sample Lectures) 69
TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin
www.mk.tu-berlin.de
Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]
Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt
Berlin, 1. Month 2014
Subject: Text…& Prof. Dr. Giuseppe Caire
End of Lecture 2
Copyright G. Caire (Sample Lectures) 70