Exercises for the seminar of the lecture Communications 2...

Exercises for the seminar of the lecture

Communications 2

Prof. Dr.-Ing. A. Czylwik

N T S

The document can contain errors inspite of careful inspections.

Tarik Akbudak, M.Sc.Fachgebiet Nachrichtentechnische SystemeRaum: BA 250, Durchwahl: -2944, eMail: [email protected]

NTS/IWUDE

Seminar exercises of the lectureCommunications 2

page1/29

Problem 1

Given a random variable x with the following probability density fx(x):

fx(x)

−1 −1

4

1

41

1

2

1

2

x

(1

2

)

Calculate the following probabilities:

a) P (x > 0)

b) P (x < 0)

c) P (x =1

4)

d) P (x = −1

4)

e) P (|x| >1

2)

Hint:

The probability that a random variable x lies within a certain interval is

P (x1 ≤ x ≤ x2) =

x2∫

x1

fx(x) dx = Fx(x2) − Fx(x1)

Date: November 10, 2010

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Solution for problem 1

a)

P (x > 0) =

1∫

0

fx(x) dx =1

2︸︷︷︸

Delta

+1

2· 1 · 1

2︸︷︷︸

Area of the triangle

=3

4

b)

P (x < 0) =

0∫

−1

fx(x) dx =1

2· 1 · 1

2︸︷︷︸


=1

4=

(

1 − P (x > 0))

c)

P (x =1

4) = lim

∆x→0

1

4+∆x∫

1

4−∆x

fx(x) dx =1

2︸︷︷︸

Weight of the delta

d)

P (x = −1

4) = lim

∆x→0

−1

4+∆x

∫

−1

4−∆x

fx(x) dx = 0

e)

P (|x| >1

2) = P (x >

1

2) + P (x < −1

2) = 2·

︸︷︷︸

d.t.t. symmetry

1∫

1

2

fx(x) dx

= 2 · 1

2· 1

2· 1

4︸︷︷︸


=1

8


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Problem 2

Given a continuous random variable x which is uniformly distributed over the interval−1 ≤ x ≤ 4.

a) Determine and sketch the probability density fx(x).

b) Sketch probability distribution Fx(x).

c) Calculate the expected value E {x} = m(1)x of the random variable x.

d) Calculate the variance σ 2x = E

{(

x − m(1)x

)2}

of the random variable x.


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a)fx(x) =

1

5·rect

(x − 3

2

5

)

x

fx(x)15

−1 4

b)

x

Fx(x)

−1 4

1

c)

mx =3

2

d)

σ 2x =

25

12


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Problem 3

Given a discrete random variable x with the probability density

fx(x) = p1 · δ(x − x1) + p2 · δ(x − x2) + p3 · δ(x − x3)

with x1 = 0, x2 = 1 and x3 = 4 and a memoryless system with the characteristic curveg(x) = 2x + 1.

a) Sketch the function g(x) and the probability density fx(x).

b) The random variable x is passed over the memoryless system with the characteristicy = g(x) and the random variable y at the output. Sketch the probability densityfy(y) and determine its formula.

c) Calculate the expected values E {x} = m(1)x and E {y} = m

(1)y of the random

variables x and y.


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a)

2 3 4 x

y = g(x)

1

1

3

5

7

9

1 2 3 4 x

fx(x)

(p2)(p1)

(p3)

13

57

9y

fy

(y)

(p3 )

(p2 )

(p1 )

b)fy(y) = p1 · δ(y − 1) + p2 · δ(y − 3) + p3 · δ(y − 9)

c)

m(1)x = p2 + 4 · p3 m

(1)y = 1 + 2 · p2 + 8 · p3 = p1 + 3 · p2 + 9 · p3


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Problem 4

Given a continuous random variable x with the probability density

fx(x) =

1

6· (x + 2) , −2 ≤ x < 0

1

3, 0 ≤ x < 2

as well as the function

y = g(x) =

{1 , −∞ < x < 0

2x + 1 , 0 ≤ x < ∞

a) Sketch the function g(x) and the probability density fx(x).

b) The random variable x is passed over the non-linear memoryless system y = g(x)with the random variable y at the output. Sketch the probability density fy(y)and determine its formula.


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a)

2 3 4 x

g(x)

1

3

5

7

9

−1−2

1 2 3 4 x−2 −1

13

57

9y

fy

(y)

fx(x)

13

(13 )

16

1

b)

fy(y) =1

3· δ(y − 1) +

1

6· rect

(y − 3

4

)


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Problem 5

In combined random experiments each outcome consists of a set of n values each associ-ated to its corresponding random variable.

For example: x = x; y = y; z = z; etc.

Thus each outcome can be considered as a distinct point in an n-dimensional coordinate-system.

Sketch the following events of a combined random experiment with two random variablesx and y using a Cartesian coordinate system x, y:

Hint:The notation {A,B} denotes the intersection of the subsets A and B, i.e. {A,B} ={A ∩ B} is the subset of all outcomes being in A and in B.

a) {x < ∞, y < ∞}

b) {x < x, y < ∞}

c) {x < ∞, y < y}

d) {x < x, y < y}

e) {x ≤ x < x + dx, y < y}

f) {x ≤ x < x + dx, y ≤ y < y + dy}

g) {x < x}

h) {x < ∞, y < k − x} with k > 0, real


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Solution for Problem 5

↑ y

x →0

↑ y

x →0

↑ y

x →0

↑ y

x →0

↑ y

x →0

↑ y

x →0

↑ y

x →0

↑ y

x →0

y = k − x

{x < x}

{x < ∞, y < ∞} {x < x, y < ∞}

{x < x, y < y}{x < ∞, y < y}

{x ≤ x < x + dx, y < y} {x ≤ x < x + dx, y ≤ y < y + dy}

{x < ∞, y < k − x} mit k > 0

d)c)

e) f)

g) h)

a) b)

x + dx

x

y y

y y + dy

x

y

x

x x + dxx


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Problem 6

Given a random variable x with the probability density

fx(x) = k · rect(

x − 2

4

)

a) Determine the constant k and sketch the probability density fx(x) of the randomvariable x.

b) Determine and sketch the probability density fz(z) of the scaled random variablez = 2 · x + 3.


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a)

k =1

4

fx(x)

1 2 3 4 5 6 7 8 9 x

14

b)

fz(z) =1

8· rect

(z − 7

8

)

9 10 11 z2 3 4 5 6 7 81

fz(z)

18


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Problem 7 (Problem 15 in exercises)

Given two statistically independent random variables x and y with probability densities

fx(x) =

0 , x < 1k1 , 1 ≤ x < 213

, 2 ≤ x < 4k1 , 4 ≤ x < 50 , x ≥ 5

and fy(y) =

0 , y < −4k2 , −4 ≤ y < −314

, −3 ≤ y < −1k2 , −1 ≤ y < 00 , y ≥ 0

a) Determine the constants k1, k2 and sketch both pdf´s.

b) The two random variables x and y are summarized to form the new random variablez = x + y.

Determine the probability density fz(z).


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Solution for problem 7 (Problem 15 in exercises)

a)

k1 =1

6and k2 =

1

4

1

31

6

1 2 3 4 5 6 −4 −3 −2 −1 1

1

21

4

fx(x) fy(y)

yx

b)

1

24

5

24

1

4

z54321−1−2−3

fz(z)


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Given the probability density fx(x) of a real random variable x.

a) Calculate the probability density fy(t)(y) of the random processes y(t) = x ·| sin(ω0t)| for ω0 > 0.

b) Assume fx(x) = rect(x − 1)

Sketch the probability density fy(t)(y) for ω0 · t =π

4.

c) What kind of probability density fy(t)(y) is obtained for ω0 · t → 0 ?


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a)

fy(y, t) =1

| sin(ω0t)|· fx

(y

| sin(ω0t)|

)

=1

| sin(ω0t)|· rect

(y − | sin(ω0t)|| sin(ω0t)|

)

b)

x →0.5 1.511√2

√2

1

y →

1√2

0

fy(t)(y)

fx(x)

c)

limω0·t→0

fy(y, t) = δ(y)


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Seminar Aufgabe 9

Given the following binary random process:

x(t) =∞∑

ν=−∞

aν · rect

t − νT − T

2T

The random variables aν with ν ∈ [−∞,∞] take the values +1 and −1 with the sameprobability. The random variables aν are statistically independent.

a) Sketch a sample function x1(t) of this process.

b) Determine the probability densities faν(aν) and fx(x, t).

c) Determine the expected value E {x(t)} = m(1)x (t) and the variance σ2

x(t).

d) Determine the autocorrelation function

Rxx(t1, t2) = E {x(t1) · x(t2)}

of the process x(t).


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a)

x1(t)

tT

1 5 7 9 11−4−7−9 3

4−10

−1

1

−6

b)

faν(aν) =

1

2· δ(aν − 1) +

1

2· δ(aν + 1)

fx(x, t) =1

2· δ(x − 1) +

1

2· δ(x + 1) 6= f(t)

c)

mx(t) = 0

σ2x(t) = 1

d)

Rxx(t1, t2) =

{1, νT ≤ t1, t2 ≤ (ν + 1)T0, else


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Problem 10

Given a stationary process n(t) with its auto-correlation function

Rnn(τ) = A · si(2π f0 τ) + B2

a) Determine the average power Pn = m(2)n = E {n2(t)} in terms of the auto-correlation

function Rnn(τ).

b) Calculate the power-density spectrum Snn(ω). Which relevance does the parame-ter f0 have?Determine the average power Pn = m

(2)n = E {n2(t)} in terms of the power-density

spectrum Snn(ω).

c) Determine the DC power PDC and the AC power PAC of the process.

d) How should the parameters A and B be chosen such that E {n(t)} = 0? Whichfundamental process is described by the resulting power-density spectrum?

Hint:

ω0 · A · si(ω0

2· t

)

b r

F2π · A · rect

(ω

ω0

)

a b r

F2π · a · δ(ω0)


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a)

Pn = m(2)n = Rnn(0) = A + B2 = σ2

n +(

m(1)n

)2

b)

Snn(ω) = F{Rnn(τ)}

=A

2f0

rect

(ω

4πf0

)

+ 2πB2δ(ω)

0 ω−2πf0 2πf0

A

2f0

(2πB2)

Snn(ω)

f0 is the cut-off frequency

Pn =1

2π

∞∫

−∞

Snn(ω) dω

=1

2π· A

2f0

· 4πf0 +1

2π· 2πB2

= A + B2

c) DC power: PDC =(

E {n(t)})2

=(

m(1)n

)2

= B2

AC power: PAC = σ2n = A

d) If B = 0, then n(t) is band-limited white noise.


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Problem 11

A comparator K (nonlinear memoryless system) is supplied by a continuous, stationaryrandom signal x(t).The transfer characteristic of the comparator can be described as follows:

y =

{1 , fur x < x0

0 , fur x ≥ x0

The limit of the comparator is denoted as x0.

a) Calculate and sketch the one-dimenisonal probability density fy(y, t) of the randomsignal at the output of the comparator as a function of an arbitrary probabilitydensity fx(x, t).It holds: P0 = P ({x ≥ x0})

b) Calculate the expected value E {y(t)} of the output signal.


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a)fy(y, t) = P0 · δ(y) + (1 − P0) · δ(y − 1)

0 1 y

P1

P0

fy(t)(y)

b)

E {y} (t) =

∫∞

−∞

y · fy(y, t)dy = (1 − P0)


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Problem 12

Given a stationary white Gaussian process x(t) with its power density Sxx(ω) = S0,located at the input of an ideal bandpass-filter with the following impulse response:

h(t) =a ωB

πsi

(ωB t

2

)

cos(ω0 t)

x(t) y(t)h(t)

a) Determine the auto-correlation function Rxx(τ) of the process x(t).

b) Determine and sketch the transfer function of the ideal bandpass-filter with allsignificant points marked on the abscissa and ordinate.

c) Determine and sketch the power-density spectrum Syy(ω) of the process y(t) withall significant points marked on the abscissa and ordinate.

d) Determine the probability density fy(y) of the process y(t).

Hint:

a · δ(t) b r

Fa

ω0 · A · si(ω0

2· t

)

b r

F2πA · rect

(ω

ω0

)

cos(ω0t) b r

Fπ · [δ (ω − ω0) − δ (ω + ω0)]

x(t) ∗ y(t) b r

FX(ω) · Y (ω)

x(t) · y(t) b r

F 1

2πX(ω) ∗ Y (ω)


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a)Rxx(τ) = S0 · δ(τ)

b)

H(ω) = a

[

rect

(ω + ω0

ωB

)

+ rect

(ω − ω0

ωB

)]

−ω0 + ωB

2−ω0 − ωB

2 ω0 + ωB

2

ω0 − ωB

2

H(ω)

ωω0−ω0

a

c)

Syy(ω) = S0 · a2

[

rect

(ω + ω0

ωB

)

+ rect

(ω − ω0

ωB

)]

−ω0 + ωB

2−ω0 − ωB

2 ω0 + ωB

2

ω0 − ωB

2

ωω0−ω0

S0 a2

Syy(ω)

d)

fy(y) =1√

2 S0 ωB aexp

(

− π y2

2 a2 S0 ωB

)


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Problem 13

Given three sample functions xi(t) (i ∈ [1, 2, 3]) of the random process x(t) and thecorresponding probabilities P (xi).

T 2T 3T 4T

T 2T 3T 4T

T 2T 3T 4T

1

x1(t)

t

1

x2(t)

t

1

x3(t)

t

P (x3) = 0, 2

P (x2) = 0, 3

P (x1) = 0, 5

a) Determine the probability density fx(x, t) of the process x(t).

b) Determine the expected value E {x(t)} = m(1)x (t) of the process x(t) and sketch it.

c) Determine the Sie die auto-correlation function Rxx(t1, t2) = E {x(t1) · x(t2)} ofthe process x(t). Sketch Rxx(t1, t2) on the t1, t2−plane.


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a)

fx(x, t) =

δ(x) , t < 0

0, 8 · δ(x − 1) + 0, 2 · δ(x), 0 ≤ t < T

δ(x − 1) , T ≤ t < 2T

0, 7 · δ(x − 1) + 0, 3 · δ(x), 2T ≤ t < 3T

0, 5 · δ(x − 1) + 0, 5 · δ(x), 3T ≤ t < 4T

δ(x) , t > 4T

b)

m(1)x (t) =

0 , t < 0

0, 8, 0 ≤ t < T

1 , T ≤ t < 2T

0, 7, 2T ≤ t < 3T

0, 5, 3T ≤ t < 4T

0 , t > 4T

t

mx(t)(t)

2 TT

0, 20, 40, 60, 8

1

c)

Rxx(t1, t2) = P (x1) · x1(t1) · x1(t2)

+P (x2) · x2(t1) · x2(t2)

+P (x3) · x3(t1) · x3(t2)


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0, 7 0, 7

0, 7

0, 5

0, 5 0, 5 0, 5 0, 5

0, 5

0, 5

0, 50, 5

1

0.8

0.8

0.8t1

2 TT

t2

T

2 T

t1

2 TT

t2

0, 5

0, 5

0, 5

0, 5 0, 5 0, 5 0, 5

0, 5 0, 5 0, 5

0, 5

0, 50, 5

0, 50, 5

0, 5

0, 30

0, 30

0, 3

0, 30

00, 2

00

00

00

0000

00

00

00

00

00

0, 20, 2

0, 2

T

2 T

Rxx(t1, t2)

P (x1) · x1(t1) · x1(t2)

P (x2) · x2(t1) · x2(t2)

P (x3) · x3(t1) · x3(t2)

2 T 2 T

3 T

2 T

T

4 T

t1T 2 T

Rxx(t1, t2)t2

0.8

0.8 0.8

1

0, 7

0, 7

0, 5 0, 5 0, 5 0, 5

0, 5

0, 5

0, 50, 5

0, 5 0, 7


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y(t) = x(t) + n(t) is the input signal of a matched filter.

The deterministic signal component x(t) is time limited in the range [0 ≤ t ≤ T ].

n(t) is white Gaussian noise with power spectral density Snn(ω) = S0.

The matched filter with impulse response h(t) = k · x(T − t) processes the signal y(t).

a) Determine the deterministic signal component gf (t) at the output.

b) Calculate the autocorrelation function Rn0n0(τ) of the noise component at the

output and show that Rn0n0(τ) can be expressed in terms of gf (τ).

c) Calculate the signal/noise ratio γa at the filter output at the sampling time t = T .

Hint: The signal/noise ratio γa is defined as follows:

γa =g2

f (T )

E {n20(T )}


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a)xf(t) = k · x(t) ∗ x(T − t); xf(T ) = k0 · Ex

b)Rn0n0

(τ) = N0 k xf(τ + T )

c)

γa =xf(T )

N0 k=

Ex

N0


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