NET 222:
COMMUNICATIONS AND
NETWORKS
FUNDAMENTALS
Tutorial 1 : Chapter 1+2 (Signals &
Systems)
Lecture 1 + Lecture 2 + Lecture 3
Networks and communications
Exercises
 1.4 (a) : Q:Let x[n] be a signal with x[n] =0 for n<-2 and n>4 , for each
signal given below, determine the values if n for which it is guaranteed to
be zero.
a) x[n-3]
Answer: The signal x[n] is shifted by 3 to the right. The shifted signal will
be zero for n <1 & n>7.
 1.5 (d) : Q: Let x(t) be a signal with x(t) =0 for t<3 , for each signal given
below, determine the values if t for which it is guaranteed to be zero.
d)x(3t)
Answer: x(3t) is obtained by linearly compressing x(t) by a factor of 3.
there for, x(3t) will by zero for t<1.
 1.21 (a) : Q: a continuous-time signal x(t) is shown in figure P1.21. sketch
and label carefully each of the following signals:
a) x(t-1)
 1.22 (a):
a discrete signal x[n] is shown in figure P1.22. sketch and label carefully
each of the following signals:
a)x[n-4]
Q1: find the value of A , f and φ from the following signals:
1
1- 𝑠(𝑡) = ( ) sin( 2 𝜋 8 𝑡 +
5
3𝜋
4
)
A= 1/5 volt ; f= 8 Hz ; φ= 3π/4
𝜋
2- 𝑠(𝑡) = (3) sin( 2 𝜋 3 𝑡 + )
3
A= 3 volt ; f= 3 Hz ; φ= π/3
Q2: find the value of spectrum and bandwidth of the following signal :
1
1- 𝑠(𝑡) = ( ) sin ( 2 𝜋 8 𝑡 +
5
3𝜋
4
𝜋
) + (3) sin( 2 𝜋 3 𝑡 + 3 ) + (4) sin( 2 𝜋 4 𝑡)
𝒔𝒑 = {3 𝐻𝑧 , 8𝐻𝑧} − 𝒃𝒘 = 8 − 3 = 5 𝐻𝑧
Q3: find the value of the following unit step functions, then draw their graphs :
1- 𝑢 [𝑛 − 5].
0 𝑖𝑓 𝑛 < 5
𝑢[𝑛 − 5] = {
1 𝑖𝑓 𝑛 ≥ 5
2- 4 𝑢 ( 𝑡 − 3).
0 𝑖𝑓 𝑡 < 3
4𝑢(𝑡 − 3) = {
4 𝑖𝑓 𝑡 ≥ 3
3- 𝑢−2 (𝑡)
0 𝑖𝑓 𝑡 < −2
𝑢−2 (𝑡) = {
1 𝑖𝑓 𝑡 ≥ −2
Q4: find the value of the following unit impulse functions, then draw their graphs :
1- 𝜕[𝑛 − 4]
𝜕[𝑛 − 4] = {
1 𝑖𝑓 𝑛 = 4
0 𝑖𝑓 𝑛 ≠ 4
2- 𝜕 ( 𝑡 + 2 )
+∞ 𝑖𝑓 𝑡 = −2
𝜕(𝑡 + 2) = {
0 𝑖𝑓 𝑡 ≠ −2
3- ∑2𝑘=−3 𝜕(𝑡 − 𝑘)
Q5: Determine whether this system are memory less, invertible, causal
Systems




𝑦(𝑡) = 3𝑥(𝑡 + 3)
𝑦(𝑡) = 𝑥(𝑡 − 2)
𝑦(𝑡) = 𝑥(5)
𝑦(𝑡) = sin(𝑥(𝑡))
Memory less
Invertible
Causal
no
no
no
yes
yes
yes
no
no
no
yes
no
yes
Q6: for the following input-output relationship, determine whether the
corresponding system is linear ,or not.
a. 𝑦[𝑛] = ∑𝑛𝑘=0 𝑥[𝑘]
b. 𝑦[𝑛] = 𝑥[𝑛]/𝑛
Q7: for the following input-output relationship, determine whether the
corresponding system is Time invariant, or not.
a. 𝑦[𝑛] = 5𝑥[𝑛 − 10]
b. 𝑦[𝑛] = 𝑥[𝑛]/𝑛
Q8:
A. An LTI system has the impulse response h[n] = { 1 , -3 , -1 , 2 }; the
underline locates the n=0 value. For the input sequences x [n] = {1 , 2 , 3 , 2
, 2 ,1}, find the output sequence y[n] = x[n] * h[n]. Express it as a list and as
a steam plot.
Y[n] = {1, -1, -4, -7, -3, -1, -1, 3, 2}
B. An LTI system has the impulse response h[n] = { 1 , 2 , 0 , -3 }; the underline
locates the n=0 value. For the input 𝑥[𝑛] = 𝛿[𝑛], find the output sequence
y[n] = x[n] * h[n]. Express it as a list and as a steam plot.
Q9: a continuous-time signal x(t) is shown in figure, sketch and write the signal
for each of the
following:
a- shift to the left by one unit along the t axis.
b- shift to the left by one unit then scaling to 2/3 along the t axis
c- shift to the left by one unit then time
reversal
Q10: Sketch the discrete-time signal x[n] is shown below 𝑥[𝑛 + 3]