1
CLASS 2
(Sections 1.3)
Exponential and Sinusoidal Signals
∙
They arise frequently in applications, and many other signals can be constructed from them.
Continuous-time complex exponential and sinusoidal signals:
𝑥(𝑡) = 𝐶𝑒𝑎𝑡
where 𝐶 and 𝑎 are in general complex numbers.
Real exponential signals: 𝐶 and 𝑎 are reals.
at
0
∙
C>0 and a<0.
Ce
Ce
at
C>0 and a>0.
C
0
t
0
C
0
t
The case 𝑎 > 0 represents exponential growth. Some signals in unstable systems exhibit exponential
growth.
∙
The case 𝑎 < 0 represents exponential decay. Some signals in stable systems exhibit exponential
decay.
2
Periodic complex exponential:
𝑒𝑗𝑤0 𝑡
where 𝑗 =
√
−1, 𝑤0 ∕= 0 is real, and 𝑡 is the time.
Euler’s formula: 𝑒𝑗𝑤0 𝑡 = cos(𝑤0 𝑡) +𝑗 sin(𝑤0 𝑡) . Note that
| {z }
| {z }
Re{𝑒𝑗𝑤0 𝑡 }
Im{𝑒𝑗𝑤0 𝑡 }
1
jw t
sin(w t)
e
Im
0
0
0
cos(w0t)
−1
−1
∙
∣𝑒𝑗𝑤0 𝑡 ∣ = 1 and ∠𝑒𝑗𝑤0 𝑡 = 𝑤0 𝑡.
∙
𝑒𝑗2𝜋𝑘 = 1, for 𝑘 = 0, ±1, ±2, . . . .
0
Re
1
Since
(
)
2𝜋
𝑗𝑤0 𝑡+ ∣𝑤
∣
𝑒
0
𝑤
𝑗2𝜋 ∣𝑤0 ∣
= 𝑒𝑗𝑤0 𝑡 𝑒
0
𝑗𝑤0 𝑡
0)
= 𝑒𝑗𝑤0 𝑡 𝑒|𝑗2𝜋sign(𝑤
{z } = 𝑒
=1
we have
𝑒𝑗𝑤0 𝑡 is periodic with fundamental period
2𝜋
.
∣𝑤0 ∣
Note that
∙
𝑒𝑗𝑤0 𝑡 and 𝑒−𝑗𝑤0 𝑡 have the same fundamental period.
∙
Energy in 𝑒𝑗𝑤0 𝑡 :
∙
Average Power in 𝑒𝑗𝑤0 𝑡 : lim𝑇 →∞
∙
{𝑒𝑗𝑘𝑤0 𝑡 }𝑘=0,±1,... , are all periodic with period
∫∞
−∞
∣𝑒𝑗𝑤0 𝑡 ∣𝑑𝑡 =
∫∞
−∞
1
2𝑇
1.𝑑𝑡 = ∞
∫𝑇
−𝑇
∣𝑒𝑗𝑤0 𝑡 ∣𝑑𝑡 = lim𝑇 →∞
2𝜋
.
∣𝑤0 ∣
1
2𝑇
∫𝑇
−𝑇
1.𝑑𝑡 = 1.
They are called a harmonically related set of
complex exponentials with 𝑒𝑗𝑘𝑤0 𝑡 being the 𝑘−th harmonic.
3
Sinusoidal signals:
𝐴 cos(𝑤0 𝑡 + 𝜙)
and
𝐴 sin(𝑤0 𝑡 + 𝜙).
where 𝐴 is real, 𝑤0 is real, 𝜙 is real, and 𝑡 is the time. (Graph one of the signals!)
∙
∙
They arise in systems that conserve energy such as an ideal LC circuit or an ideal mass-spring system.
– Periodic with the same fundamental period 𝑇0 = 2𝜋/∣𝑤0 ∣
– ∣𝑤0 ∣ is the fundamental frequency
– 𝑓0 := 1/𝑇0 = ∣𝑤0 ∣/(2𝜋) is the number of cycles per unit time (large 𝑓0 means more oscillatory)
– ∣𝐴∣ is the amplitude
– ∣𝜙∣ is the size of the phase shift.
∙
Since
𝑒𝑗(𝑤0 𝑡+𝜙) = cos(𝑤0 𝑡 + 𝜙) + 𝑗 sin(𝑤0 𝑡 + 𝜙)
we can write
𝐴 cos(𝑤0 𝑡 + 𝜙) = 𝐴Re(𝑒𝑗(𝑤0 𝑡+𝜙) )
𝐴 sin(𝑤0 𝑡 + 𝜙) = 𝐴Im(𝑒𝑗(𝑤0 𝑡+𝜙) ).
∙
Recall, for any complex number 𝑧,
𝑧 = Re(𝑧) + 𝑗Im(𝑧)
𝑧 ∗ = Re(𝑧) − 𝑗Im(𝑧)
therefore
Re(𝑧) =
𝑧 + 𝑧∗
2
Im(𝑧) =
𝑧 − 𝑧∗
.
2𝑗
Hence, we can also write
)
𝐴 ( 𝑗(𝑤0 𝑡+𝜙) ( 𝑗(𝑤0 𝑡+𝜙) )∗ ) 𝐴 ( 𝑗(𝑤0 𝑡+𝜙)
𝑒
+ 𝑒
=
𝑒
+ 𝑒−𝑗(𝑤0 𝑡+𝜙)
2
2
𝐴
𝐴
= 𝑒𝑗𝜙 𝑒𝑗𝑤0 𝑡 + 𝑒−𝑗𝜙 𝑒−𝑗𝑤0 𝑡
2
2
(
)
𝐴 𝑗(𝑤0 𝑡+𝜙) ( 𝑗(𝑤0 𝑡+𝜙) )∗ ) 𝐴 −𝑗𝜋/2 ( 𝑗(𝑤0 𝑡+𝜙)
𝐴 sin(𝑤0 𝑡 + 𝜙) =
𝑒
− 𝑒
= 𝑒
𝑒
− 𝑒−𝑗(𝑤0 𝑡+𝜙)
2𝑗
2
𝐴
𝐴
= 𝑒𝑗(𝜙−𝜋/2) 𝑒𝑗𝑤0 𝑡 − 𝑒−𝑗(𝜙+𝜋/2) 𝑒−𝑗𝑤0 𝑡 .
2
2
𝐴 cos(𝑤0 𝑡 + 𝜙) =
4
General complex exponential signals:
𝐶𝑒𝑎𝑡
where 𝐶 and 𝑎 are complex numbers.
If
𝐶 = ∣𝐶∣𝑒𝑗𝜃
and
𝑎 = 𝑟 + 𝑗𝑤0
then
𝐶𝑒𝑎𝑡 = ∣𝐶∣𝑒𝑗𝜃 𝑒(𝑟+𝑗𝑤0 )𝑡 = ∣𝐶∣𝑒𝑟𝑡 𝑒𝑗(𝑤0 𝑡+𝜃) = ∣𝐶∣𝑒𝑟𝑡 cos(𝑤0 𝑡 + 𝜃) +𝑗 ∣𝐶∣𝑒𝑟𝑡 sin(𝑤0 𝑡 + 𝜃) .
|
{z
}
|
{z
}
Re(𝐶𝑒𝑎𝑡 )
Im(𝐶𝑒𝑎𝑡 )
r>0.
r<0.
|C|ert
at
Re(Ce )
at
Re(Ce )
|C|ert
0
0
−|C|ert
0
−|C|e
0
t
rt
t
∙
If 𝑟 = 0, the real and imaginary part are sinusoidals.
∙
If 𝑟 > 0, the real and imaginary part are sinusoidals multiplied by a growing exponential.
Such signals arise in unstable systems.
∙
If 𝑟 < 0, the real and imaginary part are sinusoidals multiplied by a decaying exponential.
Such signals arise in stable systems, for example, in RLC circuits, or in mass-spring-friction system,
where the energy is dissipated due to the resistors, friction, etc.
5
Discrete-time complex exponential and sinusoidal signals:
𝑥[𝑛] = 𝐶𝑒𝛽𝑛
where 𝐶 and 𝛽 are complex numbers.
Analogous to the continuous-time case with the following differences: (𝑤0 is real below)
∙
𝑒𝑗𝑤0 𝑡 = 𝑒𝑗𝑤1 𝑡 are different signals if 𝑤0 ∕= 𝑤1 , whereas
𝑒𝑗𝑤0 𝑛 = 𝑒𝑗𝑤1 𝑛
if
𝑤0 − 𝑤1 = 2𝑘𝜋, for some 𝑘 ∈ {0, ±1, . . . }.
(Explain this on the unit circle!)
Therefore, it is sufficient to consider only the case 𝑤0 ∈ [0, 2𝜋) or 𝑤0 ∈ [−𝜋, 𝜋).
∙
As 𝑤0 increases 𝑒𝑗𝑤0 𝑛 oscillates at higher frequencies, whereas this is not the case for 𝑒𝑗𝑤0 𝑛 .
In the figure below, the frequency of oscillations increases as 𝑤0 changes from 0 to 𝜋 then it decreases
as 𝑤0 changes from 𝜋 to 2𝜋.
∙
𝑒𝑗𝑤0 𝑡 is periodic with fundamental period 2𝜋/∣𝑤0 ∣, whereas
𝑒𝑗𝑤0 𝑛 is periodic
⇔ 𝑒𝑗𝑤0 𝑛 = 𝑒𝑗𝑤0 (𝑛+𝑀 ) for some integer 𝑀 > 0, for all 𝑛
⇔ 𝑒𝑗𝑤0 𝑀 = 1 for some integer 𝑀 > 0
⇔ 𝑤0 𝑀 = 2𝜋𝑚 for some integers 𝑚, 𝑀 > 0
⇔
∙
If
𝑤0
2𝜋
=
is 𝑀 =
𝑚
𝑀
𝑤0
is rational.
2𝜋
for some integers 𝑚 and 𝑀 which have no common factors, then the fundamental period
2𝑚𝜋
𝑤0
because
𝑒𝑗𝑤0 (𝑛+𝑁 ) = 𝑒𝑗𝑤0 𝑛 𝑒𝑗
The same observations hold for discrete-time sinusoids.
2𝜋𝑚
𝑁
𝑀
.
6
w =1π/8
w =0π/8
0
n=4
n=5
n=3
n=6
n=2
n=7
n=1
n=8
n=0
n=9
n=15
n=10 n=14
n=11
n=13
n=12
n=3
Im
n=0
Im
Im
w =2π/8
0
0
n=4
n=6
n=7
Re
Re
w =4π/8
w =8π/8
w =12π/8
0
0
n=2
n=0
n=1
n=0
Im
n=3
Im
Im
n=0
n=5
n=1
n=3
n=2
n=0
n=1
Re
Re
w0=14π/8
w0=15π/8
w0=16π/8
n=4
n=7
n=0
n=3
n=2
Re
n=1
Im
n=6
n=12
n=11
n=13
n=10 n=14
n=9
n=15
n=8
n=0
n=7
n=1
n=6
n=2
n=5
n=3
n=4
Im
Re
n=5
Im
n=1
Re
0
Fig. 1.
n=2
Re
To determine the fundamental period, count the number of steps to get back to 1!
n=0
Re
7
Examples:
1) Is 𝑥[𝑛] = 𝑒𝑗𝑛2𝜋/3 + 𝑒𝑗𝑛3𝜋/4 periodic? If it is periodic, what’s its fundamental period?
For 𝑒𝑗𝑛2𝜋/3 , 𝑤0 /(2𝜋) = 1/3, so 𝑒𝑗𝑛2𝜋/3 is periodic with fundamental period 3.
For 𝑒𝑗𝑛3𝜋/4 , 𝑤0 /(2𝜋) = 3/8, so 𝑒𝑗𝑛3𝜋/4 is periodic with fundamental period 8.
𝑥[𝑛] is periodic with fundamental period 24 = 𝑙𝑐𝑚(3, 8).
2) Is 𝑥[𝑛] = sin(3𝑛/4) periodic? If it is periodic, what’s its fundamental period?
Since
𝑤0
2𝜋
=
3
8𝜋
is irrational, 𝑥[𝑛] is not periodic; see the figure where 𝑥[𝑛] = 0 only at 𝑛 = 0.
sin(3t/4) and sin(3n/4)
1
0
−1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
t
3) Is 𝑥[𝑛] = sin(8𝜋𝑛/31) periodic? If it is periodic, what’s its fundamental period?
Since 𝑤0 /(2𝜋) = 4/31, 𝑥[𝑛] is periodic with fundamental period 31; see the figure where 𝑥[0] =
𝑥[31] = 0. Note that the continuous-time signal sin(8𝜋𝑡/31) has fundamental period 31/4, hence it
is 0 at 𝑡 = 31/4. But 𝑥[𝑛] has no 31/4−th sample and it misses 0 between 𝑥[7] and 𝑥[8].
sin(8π t/31) and sin(8π n/31)
1
0
−1
0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435
t
8
Harmonically related discrete-time periodic exponentials:
𝜙𝑘 [𝑛] = {𝑒𝑗𝑘(2𝜋/𝑁 )𝑛 }𝑘=0,±1,... , are all periodic with period 𝑁 .
However, unlike the continuous-time signals, these signals are not all distinct because
𝜙𝑘+𝑁 [𝑛] = 𝑒𝑗(𝑘+𝑁 )(2𝜋/𝑁 )𝑛 = 𝑒𝑗𝑘(2𝜋/𝑁 )𝑛 𝑒𝑗2𝜋𝑛 = 𝜙𝑘 [𝑛].
This implies that there are only 𝑁 distinct signals in this set, for example,
𝜙0 [𝑛] = 1
𝜙1 [𝑛] = 𝑒𝑗2𝜋𝑛/𝑁
𝜙2 [𝑛] = 𝑒𝑗4𝜋𝑛/𝑁
..
.
𝜙1 [𝑛] = 𝑒𝑗2(𝑁 −1)𝜋/𝑁 .