2.6 Post-Reading and Exercises

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18. Generate another signal analogous to the previous task by using the cosine func-

tion. What covariance of the two signals do you expect? Check your assumption

with the help of the Matlab-function cov(). Now replace the cosine function with

a sine function and discuss the result.

19. Explain the difference between analogue and digital signals using the technical

terms. What is the main advantage of processing digital signals? Using Matlab,

plot a sine function as a continuous or discrete signal following Figure 2.19 using

the functions provided.

Transformations of Signal Processing

1.

What is the mathematical form of the Fourier series and what is it used for? How

is it different from the Fourier transform?

2.

What are the coefficients of the Fourier series? What is their significance in fre-

quency analysis?

3.

What is the Gibbs phenomenon, when does it occur in the representation of sig-

nals using the Fourier series? Can it be avoided?

4.

Describe the consequences of the individual theoremsoftheFourier-transformation,

how do the Fourier pairs of linearity, time/frequency shift, multiplication and con-

volution have an effect in the time and frequency domain respectively? Interpret.

5.

What fundamental postulate underlies the process of spectral analysis? Without

this assumption, spectral analysis would be meaningless.

6.

Describe the behaviour of a linear time-invariant system in the time and frequency

domain with the help of the Fourier-transformation. What are the computational

advantages in the individual forms of representation with the aid of the Fourier

theorems?

7.

Describe the integral transformations in general, what is the significance of the

integral kernel? Name the most important integral transformations in signal pro-

cessing and their integral kernels. What are the differences? Interpret.

8.

What form does a sine or cosine transformation take? What results do you expect

in comparison to the Fourier transformation for exclusively odd or even signals?

9.

Under which conditions does a solution of the Fourier-transformation exist, in

which cases does it not converge? Do you know a practicable solution to transform

non-converging signals in another way?

10. Describe the significance of convolution in relation to linear, time-invariant sys-

tems of signal processing.

11. Explain the mathematical approach to convolution, how can its result be inter-

preted? Carry out the convolution of two rectangular functions as an example,

interpret the convolution product. What does the convolution of a function with

the Dirac momentum look like?