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 a0 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, 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 .