Question 1 EEE30004 Digital Signal Processing Tutorial Question Sheet 1 Sinusoids The phase of a sinusoid can be related to time shift: x(+) = A cos(2 fot +() = A cos(2x fo(1 -t1) In the following parts, assume that the period of the sinusoidal wave is T = 10 sec. (a) "When t1 = - 2 sec, the value of the phase is ¢ = x /5." Explain whether this is TRUE or FALSE. (b) "When +1 = 5 sec, the value of the phase is ø = x." Explain whether this is TRUE or FALSE. (c) "When 11 = 8 sec, the value of the phase is ø = 2x /5." Explain whether this is TRUE or FALSE. Question 2 The phase of a sinusoid x(1) = A cos(2x fot ++) can be related to the time shift of a positive peak. (a) When the frequency is f. = 10 Hz and the time of a positive maximum is tm = - 0.04 sec, determine the value for the phase ¢ (in radians). (b) Now assume that the frequency f. is unknown. If the phase is ø = +x/2 radians and x(t) has one of its positive peaks at tm = 3 sec, determine the smallest value for the frequency f. (in Hz). Make sure that your answer for f. is positive. (c) The answer in part (b) was the smallest one, but there are many other solutions that also have ø = +x/2 rads. and tm = 3 sec. Give a general formula for all possible frequencies. Question 3 Let x [n] be the complex exponential x[n] = ej(0.47n-0.5?) If we define a new signal y[n] to be the first difference: y[n] = x[n] - x[n - 1], it is possible to express y[n] in the form y[n]= Ae(won+) Determine the numerical values of A, ¢ and wo. (Should wo be equal to 0.4x?) Question 4 In all parts of this problem, consider a signal x[n] = 20cos(0.5mn +?). (a) The signal x[n] can be represented as x[n] = Re{XejWon}. Determine X and wo and plot X as a vector in the complex plane. (b) Consider the signal w[n] = >[n - 5] which can be expressed as w[n] = Re{Wejwon}. What operation on the phasor X corresponds to the the operation of time delay by 5 samples? That is, how is W related to X? Express your answer both in terms of the general symbol @o and in terms of the numerical value of @o determined in part (a). (c) Express the signal y[n] = x[n]+w[n] in the form y[n] = A cos(@on+p). Plot in the complex plane, all the phasors used in the solution.
Question 5 Simplify the following and give the answer as a single sinusoid. (a) xa(1) = 2cos(7?? -3?/4) + cos (7?t +?/2) (b) xb(t)=cos(33x++3/2)+3cos(33xt+6x) (c) xc(1)= cos(1+3?/2)+cos(?? + 5?/6) +cos (?? + ?/6)
Question 1 EEE30004 Digital Signal Processing Tutorial Question Sheet 2 Spectrum Representation The spectrum of a signal x(1) is