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2 Fundamentals of Information, Signal and System Theory

4.

Draw an arbitrarily shaped, causal, aperiodic time signal in a labelled Cartesian

coordinate system.

5.

What is the relationship between the period and the frequency of a sinusoidal

signal?

6.

Interpret the formula of the Fourier series with regard to the postulate mentioned

earlier that every signal is built up from a sum of harmonic signals of different

amplitude and frequency.

7.

Explain the term linear superposition using a sawtooth oscillation.

8.

Use the superposition of two sine functions to produce a signal similar to the one

in Figure 2.11 (bottom right). Alternatively, use the Fourier series. What is the effect

of changing the amplitude and phase of each harmonic?

9.

What special properties does the delta-distribution have, how can these be ex-

pressed mathematically?

10. How can a DC-function in the time domain 1(t) also be described mathematically?

To do this, use a suitable limit analysis for the period of a complex exponential

function.

11. Explain clearly, using a limit value consideration of the period, why the sampling

function (Dirac-pulse train) again has a sampling function as a Fourier-transform,

although the Fourier-transform of the delta-function is a DC-function.

12. Analogous to Listing 2.3.2 create a Matlab-script for the graphical representation

of the two functions from Figure 2.13.

(a) Change the sign in the exponent of the exponential function to create a tran-

sient.

(b) Gradually decrease the value of the standard deviation of the normal distribu-

tion and perform the limit transition to the Dirac-distribution to some extent.

To do this, use Equation 2.21 and compare the output with Figure 2.14.

13. Describe the sieve property of the Dirac-distribution and how it acts on a signal

f(t). What is the effect of shifting the Dirac-distribution and how is this done in

the positive time direction?

14. What is the relationship between the delta-distributionandtheHeaviside-function?

How do you explain the connection and the discontinuity with the help of the de-

rivative?

15. Name the properties for energy and power signals with an example for each.

Check by calculation whether the square wave pulse, the Dirac pulse, the sine

function, the exponential function and the DC signal are energy or power signals.

16. Explain the different terms of stationarity of a stochastic signal and give examples

for the respective category.

17.

Generate a mixed signal from a deterministic sine function and a noise signal in

an additive way, analogous to the representation in Figure 2.17 (right). To do this,

use the Matlab-functions mean(), var() and std() to calculate the respective mag-

nitudes of the first two moments.