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1. Chapter One
Discrete-Time Signals and Systems
Lecture #2
Rediet Million
AAiT, School Of Electrical and Computer Engineering
rediet.million@aait.edu.et
March,2018
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2. 1.2.Discrete-Time Signals and Systems
1.2.1. Discrete-Time Signals
A discrete-time signal is a function of a discrete time represented as a
sequence of number x(n) :
X = {x(n)} − ∞ < n < ∞
x(n) is the nth sample of the sequence.
x(n) is only defined for n an integer.
Discrete time signal often obtained by periodic sampling of a
continuous-time signal.
Sequence x(n) = xa(nT) ,n = −1, 0, 1, 2.3... where T is the
sampling period.
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3. Discrete-time Signals
Sequences
Discrete-time signals may be represented mathematically in several
ways:
i) Functional representation:
x(n) =
(0.5)n , n ≥ 0
0 , n < 0
ii) Sequence representation:
x(n) = {..., −0.5, 1.5, 3, −3.7, 2.9, 0.6...}
↑
- The arrow indicates where n=0.
- Thus ,x(−1) = −0.5, x(0) = 1.5, x(1) = 3, x()] = −3.7...
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4. Discrete-time Signals
Sequence Operations
Summation:Point-wise addition of signals expressed as
y(n) = x1(n) + x2(n)
Multiplication: Point-wise multiplication of signals
y(n) = x1(n)x2(n)
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5. Discrete-time Signals
Sequence Operations
Time shifting: y(n) = x(n − N) where N is an integer.
if N > 0 , it is delaying operation.
if N < 0 , it is an advance operation.
Unit delay
y(n) = x(n − 1)
Unit advance
y(n) = x(n + 1)
Example:When N = 2, x(n − 2) is a said to be a delayed version of x(n)
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6. Discrete-time Signals
Basic Sequences
General(Arbitrary)sequence:
Its random and not suitable for computation.Thus, we have to
represent using basic sequences.
Basic sequence :Unit samples(Impulse)sequence, Unit step
Sequence, Exponential sequence, complex sinusoidal sequence.
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7. Discrete-Time Signals
Basic Sequences
(i) Unit sample sequence: Defined as
δ(n) =
1 , n = 0
0 , n = 0
Any arbitrary sequence can be represented as a sum of scaled,delayed
and advanced unit sample sequences.
x(n) = x(−1)δ(n + 1) + x(0)δ(n) + x(1)δ(n − 1) + x(2)δ(n − 2) + .....
More generally
x[n] =
∞
k=−∞
x[k]δ[n − k]
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8. Discrete-Time Signals
Basic Sequences
(ii) Unit step sequence: Defined as
u(n) =
1 , n ≥ 0
0 , n < 0
Related to the unit sample(impulse) by .
u(n) = δ(n) + δ(n − 1) + δ(n − 2) + .....
or
u[n] =
∞
k=−∞
δ[n − k]
Conversely, δ[n] = u[n] − u[n − 1]
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9. Discrete-Time Signals |Basic Sequences
(#1) Class exercises & Assignment
1) A sequence x(n) is shown below.Express x(n) as a linear combination
of weighted and delayed unit samples.
2) Using the above sequence x(n),determine and plot the following
related sequences.
a.x(n − 2) c.x(2n) a.x(n + 2) c.x(n/2)
b.x(−n) d.δ(x(n)) b.x(3 − n) d. x(n)=δ[sin(
πn
4
)]
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10. Discrete-Time Signals |Basic Sequences
(iii) Exponential sequence
Extremely important in representing and analyzing LTI systems.
The general form of an exponential sequence is :
x(n) = αn
, for all n.
Where the parameter α is a real or complex.
When ’α’ is real, the exponential sequence takes either of the
following four forms:
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11. Discrete-Time Signals |Basic Sequences
When ’α’ is complex, a more general case to consider is the complex
exponential sequence :
x(n) = Aαn
, where α = |α|ejωo
and A = |A|ejφ
x(n) = Aαn
=|A|ejφ
|α|n
ejωon
= |A||α|n
ej(ωon+φ)
– polar form
= |A||α|n
cos(ωon + φ) + j|A||α|n
sin(ωon + φ)
= Re{x(n)} + jIm{x(n)} i.e rectangular representation of x(n)
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12. Discrete-Time Signals |Basic Sequences
if |α| < 1, the real and imaginary part of the sequence magnitude
oscillate with exponentially decaying envelopes.
if |α| > 1, the real and imaginary part of the sequence magnitude
oscillate with exponentially growing envelopes.
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13. Discrete-Time Signals |Basic Sequences
When |α| = 1, x(n) is referred to as the discrete-time complex
sinusoidal sequence and has the form:
x(n) = |A|cos(ωon + φ) + j|A|sin(ωon + φ)
For complex sinusoidal sequence, the real and imaginary part of the
sequence magnitude oscillate with constant envelopes.
The quantity ωo is called the frequency of the complex exponential or
sinusoid and φ is called the phase.
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14. Discrete-Time Signals |Basic Sequences
Frequency range: Complex exponential sequences having frequencies
ωo and ωo + 2πr,r an integer,are indistinguishable from one another.
Consider a frequency (ωo + 2π)
x(n) = Aej(ωo+2π)n
= Aejωon
ej2πn
= Aejωon
More generally (ωo + 2πr), r being an integer ,
x(n) = Aej(ωo+2πr)n
= Aejωon
ej2πnr
= Aejωon
The same for sinusoidal sequences
x(n) = Acos[(ωo + 2πr)n + φ] = Acos(ωon + φ)
This implies that a complete description of x(n) is obtained by
considering frequencies in the range of length 2π such as
−π < ωo ≤ π or 0 < ωo ≤ 2π
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15. Discrete-Time Signals |Basic Sequences
periodicity:
In the continuous-time case, a sinusoidal signal and a complex
exponential signal are both periodic.
In the discrete-time case, a periodic sequence is defined as
x(n) = x(n + N) , for all n
where the Period N is necessarily to be an integer.
For sinusoidal sequences
Acos(ωon + φ) = Acos(ωon + ωoN + φ)
which requires that ωoN = 2πr or N = 2πr
ωo
where r is an integer.
This holds the same for complex exponential sequence
ejωo(n+N) = ejωon which is true only for ωoN = 2πr
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16. Discrete-Time Signals |Basic Sequences
So,complex exponential and sinusoidal sequences
- are not necessarily periodic in n with period (2π
ωo
)
- and ,depending on the value of ωo, may not be periodic at all.
for example, for ωo = 3π
4 the smallest value of N is 8,obtained with
k=3.But,x(n) cannot be periodic for ω = 1 ,since N = 2πr can never be
an integer.
Concept of low and high frequencies:
For a continuous-time sinusoidal signal x(t) = Acos(Ωot + φ), as
Ωo increases, x(t) oscillates more and more rapidly.
The frequency of a discrete-time sinusoidal signal
x(n) = Acos(ωon + φ)
-increase as ωo increases,from ω = 0 to ω = π
-and,decreases as ωo increases,from ω = π to ω = 2π
The value of ωo in the vicinity of ωo = 2πr –> low frequencies,
the value of ωo in the vicinity of ωo = (π + 2πr) –> high frequencies
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17. Discrete-Time Signals |Basic Sequences
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18. Discrete-Time Signals | Basic Sequences
Reading assignment
Classification of Discrete-Time Signals.
Energy Signals and Power Signals
Periodic Signals and Aperiodic Signals
Symmetric and Anti-symmetric Signals
( From textbook page 11-15 )
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