School of Electrical, Computer and Telecommunications Engineering, University of Wollongong

Advanced Signals & Systems (ECTE906) Spring - Week02 Tut Questions: Review: Sinusoids & Complex Exponentials --------------------------------------------------------------------------------------------------2-1- (review-week1) There are several classes of signals. These could include; continuous-time and discrete-time signals, analog and digital signals, periodic and aperiodic signals and energy and power signals. Provide a short description for each class and demonstrate (sketch) signals for analog-continuous time, digital-continuous time, analog discrete time and digital discrete time. 2-2- What is the main advantage of a periodic signal? 2-3- One of the reasons that cosine waves are so important is that many physical systems generate signals that can be represented mathematically as sine or cosine functions of time. Among the most prominent of these are signals that are audible to humans. Provide two examples for physical systems that generate audible sound to humans. 2-4- The process of converting an analogue signal to digital signal two terms are frequently used; resolution and quantizing. Very briefly explain these terms. 2-5- The higher the resolution (number of bits) in analogue to digital conversion, the more accurate the measurement. An 8-bit ADC offers 256 discrete levels and a 12-bit ADC offers 4,096 discrete levels. If 10V applied to an 8-bit and a 12-bit converter what would be the smallest voltage range that these converters can deliver? 2-6- Not all frequencies are audible. The hi-fi range is defined as from 20 Hz to 20 kHz, approximately the same as the range of frequencies which can normally be detected. The "pitch" and "intensity" or "loudness" of a musical note corresponds (approximately) to which parameters of a sine signal that we know? 2-7- There are some useful signal operations. These include “shifting”, “scaling” and “inversion”. Since the independent variable in common signal description is time, these operations are called “time-shifting”, “time-scaling” and “time-reversal (inversion>”. Very briefly explain the general concept of “time-shifting” of a signal accompanied with the sketch of a sample original and time-shifted signal. 2-8- In electrical signals such as electric current and voltage, it is common to refer to cosine and sine signals as “waves” (or “waveforms”). What are the common names of these signals in acoustic, signal processing and signals and systems disciplines? 2-9- Complex numbers can be represented as points in a “complex plain”. Two methods are available to represent complex numbers in a complex plain; rectangular (Cartesian) and polar notation (forms). For elementary manipulations of complex numbers these notations are fine. They, however, do not lend themselves to ordinary algebraic rules. Very briefly explain polar and rectangular notations for elementary manipulations of complex numbers (addition, multiplication, division and subtraction) and outline the significance of Euler’s formula for representation of complex numbers. Your answer you be accompanied with examples both manual and MATLAB solutions.

P Doulai - Spring 2009

Tutorial Questions

School of Electrical, Computer and Telecommunications Engineering, University of Wollongong

2-10- Consider the following two sinusoids: x1 (t ) = 5 cos(2π (100)t + π / 3) abcd-

x 2 (t ) = 4 cos(2π (100)t − π / 4) Use MATLAB to plot these two signals and their sum Obtain the phasor representation of these two signals and the resulting phasor from their addition. Plot the two phasors and their sum in the complex plane. Show that the sum of the two signals is equal to: x 3 (t ) = 5.536 cos(2π (100)t + 0.2747)

2-11- Use complex exponentials (i.e. phasors) to show the following trigonometric identities: (a) (b) 2-12 - Define x(t) as Simplify x(t) into the standard sinusoidal form: . Use phasors to do the algebra, but also provide a plot of the vectors representing each of the three phasors. 2-13 - Solve the following equations for

:

2-14 - Define x(t) as

Express x(t) in the form . Use complex phasor manipulations to obtain the answer. Explain your answer by giving a phasor diagram. 2-15 - Consider the sinusoidal signals x(t) and y(t) plotted below. (a) Determine: A, Ф, and f0

P Doulai - Spring 2009

Tutorial Questions