Problems 5058.10. Periodicity of discrete-time signals
Consider periodic signalsx[n], of periodN 1 = 4 , andy[n], of periodN 2 = 6. What would be the period of
z[n]=x[n]+y[n]
v[n]=x[n]y[n]
w[n]=x[2n]8.11. Periodicity of sum and product of periodic signals—MATLAB
Ifx[n]is periodic of periodN 1 > 0 andy[n]is periodic of periodN 2 > 0 :
(a) What should be the condition for the sum ofx[n]andy[n]to be periodic?
(b) What would be the period of the productx[n]y[n]?
(c) Would the formula
N 1 N 2
gcd(N1,N 2 )
(gcd(N 1 ,N 2 )stands for the greatest common divisor ofN 1 andN 2 ) give the period of the sum and the
product of the two signalsx[n]andy[n]?
(d) Use MATLAB to plot the signalsx[n]=cos( 2 πn/ 3 )u[n], andy[n]=( 1 +sin( 6 πn/ 7 ))u[n], their sum
and product, and to find their periods and to verify your analytic results.
8.12. Echoing of music—MATLAB
An effect similar to multipath in acoustics is echoing or reverberation. To see the effects of an echo in an
acoustic signal consider the simulation of echoes on the “handel.mat” signaly[n]. Pretend that this piece is
being played in a round theater where the orchestra is in the middle of two concentric circles and the walls
on one half side are at a radial distances of 17.15 meters (corresponding to the inner circle) and 34.3 meters
(corresponding to the outer circle) on the other side (yes, an usual theater!) from the orchestra. The speed
of sound is 343 meters/sec. Assume that the recorded signal is the sum of the original signaly[n]and
attenuated echoes from the two walls so that the recorded signal is given by
r[n]=y[n]+0.8y[n−N 1 ]+0.6y[n−N 2 ]whereN 1 is the delay caused by the closest wall andN 2 is the delay caused by the farther wall. The
recorder is at the center of the auditorium where the orchestra is and we record for 1.5 seconds.
(a) Find the values of the two delaysN 1 andN 2. Give the expression for the discrete-time recorded signal
r[n]. The sampling frequencyFsof “handel.mat” is given when you load it in MATLAB.
(b) Simulate the echo signal. Plotr[n]. Usesoundto listen to the original and the echoed signals.
8.13. Envelope modulation—MATLAB
In the generation of music by computer, the process of modulation is extremely important. When playing
an instrument, the player typically does it in three stages: (1) rise time or attack, (2) sustained time, and
(3) decay time. Suppose we model these three stages as an envelope continuous-time signal given by
e(t)=
1
3
[r(t)−r(t− 3 )]−
1
0.1
[r(t− 20 )+r(t− 30 )]wherer(t)is the ramp signal.
(a) For a simple tonex(t)=cos( 2 π/T 0 t), the modulated signal isy(t)=x(t)e(t). Find the periodT 0 so that
100 cycles of the sinusoid occur for the duration of the envelope signal.
(b) Simulate in MATLAB the modulated signal using the value ofT 0 = 1 and a simulation sampling time
of0.1T 0. Ploty(t)ande(t)(discretized with the sampling period0.1T 0 ) and listen to the modulated
signal usingsound.