5.4 Post-Reading and Exercises

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6.

To what number is a voltage value of 2 V at the input of a 10-bit A/D-converter with

an input span of 0 V–10 V mapped? Let the output number range start at zero and

cross over only positive numbers. Note that this is a unipolar system.

7.

A continuous-time signal from a sensor is to be transmitted via a digital transmis-

sion system. The spectral components of the sensor that are important for analysis

are below 100 kHz. Frequency components uninteresting for the analysis extend

up to 1 MHz. a) Which frequency should be sampled? Sketch the spectrum of the

sampled signal! Is an "anti-aliasing" filter required? b) Assuming that each sample

is encoded with 8 bits, what is the minimum data rate that the transmission sys-

tem must provide (in kBit/s)?

8.

An input signal with two superimposed sinusoidal signals f1 = 3 kHz and f2 =

6 kHz is sampled with a sampling frequency fa = 8 kHz. Draw the spectrum of the

input signal before and after sampling. What does this mean for a reconstruction

of the original signal?

9.

How do you recover a continuous signal from a discrete-time one?

Discrete Transformations

1.

What are the special features of spectra of sampled signals?

2.

What is meant by the first Nyquist range of a spectrum?

3.

What is the magnitude of the highest frequency that can be represented in a spec-

trum, and what is the magnitude of its frequency resolution?

4.

What is the leakage effect of a spectrum?

5.

What is the difference between a magnitude spectrum and an amplitude spec-

trum? What is a phase spectrum? How are these calculated from the Fourier-

coefficients?

6.

What is the Fourier-transform of a square wave function?

7.

What condition must a function fulfil in order to be able to develop it using a Four-

ier series? How can this function be described?

8.

What is windowing of a signal? What is it used for?

9.

Given is the sequence of a rectangular pulse x[n] = 1, 1, 1, 1, 1, 1. Sketch the

sequence and its even and odd parts.

10. Given the two sequences x1[n] = 1, 2, 1 and x2[n] = 1, 0, 1, 0, 1, 0, 1. a) Draw

the sequences and give their lengths. b) Fold the sequences, x1[n] ∗x2[n] = y[n],

and graph the result.

11. Besides the DFT, why do we need another transformation, the z-transformation?

12. What is the shift operator of the z- transform in the z- domain?

13. A sampled weight sequence of Dirac pulses (sampling sequence, discrete time

series) is to be z- transformed. How can the shift operator be used for this?