CONTINUOUS TIME FOURIER SERIES CHAPTER 3.3-3.8 16 CTFS TRANSFORM - - PowerPoint PPT Presentation

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15 CONTINUOUS TIME FOURIER SERIES CHAPTER 3.3-3.8 16 CTFS TRANSFORM PAIR Suppose () can be expressed as a linear combination of harmonic complex exponentials 0 synthesis equation

SLIDE 1

CONTINUOUS TIME FOURIER SERIES

CHAPTER 3.3-3.8 15

SLIDE 2

CTFS TRANSFORM PAIR

 Suppose 𝑦(𝑢) can be expressed as a linear combination of

harmonic complex exponentials

 𝑦 𝑢 = σ𝑙=−∞

∞

𝑏𝑙𝑓𝑘𝑙𝜕0𝑢 synthesis equation  Then the FS coefficients {𝑏𝑙} can be found as

 𝑏𝑙 = 1

𝑈 ∫ 𝑈 𝑦(𝑢) 𝑓−𝑘𝑙𝜕0𝑢𝑒𝑢

analysis equation  𝜕0 - fundamental frequency  𝑈 = 2𝜌/𝜕0 - fundamental period  𝑏𝑙 known as FS coefficients or spectral coefficients

SLIDE 3

CTFS PROOF

 While we can prove this, it is not well suited for

slides.

 See additional handout for details

 Key observation from proof: Complex exponentials

are orthogonal

SLIDE 4

VECTOR SPACE OF PERIODIC SIGNALS

All signals Periodic signals, 𝜕0

SLIDE 5

 Each of the harmonic

exponentials are orthogonal to each other and span the space

f periodic signals

 The projection of 𝑦(𝑢) onto a

particular harmonic (𝑏𝑙) gives the contribution of that complex exponential to building 𝑦 𝑢

 𝑏𝑙 is how much of each harmonic

is required to construct the periodic signal 𝑦(𝑢)

VECTOR SPACE OF PERIODIC SIGNALS

Periodic signals, 𝜕0 𝑦(𝑢) 𝑓𝑘0𝑢 = 1 𝑏0 𝑓𝑘(−𝜕0)𝑢 𝑓𝑘𝜕0𝑢 𝑓𝑘2𝜕0𝑢 𝑓𝑘𝑙𝜕0𝑢 𝑏−1 𝑏1 𝑏2 𝑏𝑙

SLIDE 6

HARMONICS

 𝑙 = ±1 ⇒ fundamental component (first harmonic)

 Frequency 𝜕0, period 𝑈 = 2𝜌/𝜕0

 𝑙 = ±2 ⇒ second harmonic

 Frequency 𝜕2 = 2𝜕0, period 𝑈2 = 𝑈/2 (half period)

 …  𝑙 = ±𝑂 ⇒ Nth harmonic

 Frequency 𝜕𝑂 = 𝑂𝜕0, period 𝑈𝑂 = 𝑈/𝑂 (1/N period)

 𝑙 = 0 ⇒ 𝑏0 =

1 𝑈 ∫ 𝑈 𝑦 𝑢 𝑒𝑢, DC, constant component, average

ver a single period

SLIDE 7

HOW TO FIND FS REPRESENTATION

 Will use important examples to demonstrate

common techniques

 Sinusoidal signals – Euler’s relationship  Direct FS integral evaluation  FS properties table and transform pairs

SLIDE 8

 𝑦 𝑢 = 1 +

1 2 cos 2𝜌𝑢 + sin 3𝜌𝑢

 First find the period



Constant 1 has arbitrary period



cos 2𝜌𝑢 has period 𝑈

1 = 1



sin 3𝜌𝑢 has period 𝑈2 = 2/3



𝑈 = 2, 𝜕0 = 2𝜌/𝑈 = 𝜌  Rewrite 𝑦 𝑢 using Euler’s and read off 𝑏𝑙

coefficients by inspection



𝑦 𝑢 = 1 +

1 4 𝑓𝑘2𝜕0𝑢 + 𝑓−𝑘2𝜕0𝑢 + 1 2𝑘 𝑓𝑘3𝜕0𝑢 − 𝑓−𝑘3𝜕0𝑢



Read off coeff. directly



𝑏0 = 1



𝑏1 = 𝑏−1 = 0



𝑏2 = 𝑏−2 = 1/4



𝑏3 = 1/2𝑘, 𝑏−3 = −1/2𝑘



𝑏𝑙 = 0, else

SINUSOIDAL SIGNAL

SLIDE 9

 𝑦 𝑢 = ቐ

1 𝑢 < 𝑈

𝑈

1 < 𝑢 < 𝑈 2

PERIODIC RECTANGLE WAVE

SLIDE 10

 Important signal/function in

DSP and communication

 sinc 𝑦 =

sin 𝜌𝑦 𝜌𝑦

normalized

 sinc 𝑦 =

sin 𝑦 𝑦

unnormalized

 Modulated sine function

 Amplitude follows 1/x  Must use L’Hopital’s rule to get

x=0 time

SINC FUNCTION

SLIDE 11

 Consider different “duty cycle” for

the rectangle wave

 𝑈 = 4𝑈

1 50% (square wave)

 𝑈 = 8𝑈

1 25%

 𝑈 = 16𝑈

1 12.5%

 Note all plots are still a sinc

shape

 Difference is how the sync is sampled  Longer in time (larger T) smaller

spacing in frequency  more samples between zero crossings

RECTANGLE WAVE COEFFICIENTS

SLIDE 12

 Special case of rectangle wave

with 𝑈 = 4𝑈

1  One sample between zero-crossing

 𝑏𝑙 = ቐ

1/2 𝑙 = 0

sin(𝑙𝜌/2) 𝑙𝜌

𝑓𝑚𝑡𝑓

SQUARE WAVE

SLIDE 13

 𝑦 𝑢 = σ𝑙=−∞

∞

𝜀(𝑢 − 𝑙𝑈)

 Using FS integral



Notice only one impulse in the interval 27

PERIODIC IMPULSE TRAIN

CONTINUOUS TIME FOURIER SERIES

CTFS TRANSFORM PAIR

 Suppose 𝑦(𝑢) can be expressed as a linear combination of

harmonic complex exponentials

 𝑦 𝑢 = σ𝑙=−∞

∞

𝑏𝑙𝑓𝑘𝑙𝜕0𝑢 synthesis equation  Then the FS coefficients {𝑏𝑙} can be found as

 𝑏𝑙 = 1

𝑈 ∫ 𝑈 𝑦(𝑢) 𝑓−𝑘𝑙𝜕0𝑢𝑒𝑢

analysis equation  𝜕0 - fundamental frequency  𝑈 = 2𝜌/𝜕0 - fundamental period  𝑏𝑙 known as FS coefficients or spectral coefficients

CTFS PROOF

 While we can prove this, it is not well suited for

slides.

 See additional handout for details

 Key observation from proof: Complex exponentials

are orthogonal

VECTOR SPACE OF PERIODIC SIGNALS

 Each of the harmonic

exponentials are orthogonal to each other and span the space

 The projection of 𝑦(𝑢) onto a

particular harmonic (𝑏𝑙) gives the contribution of that complex exponential to building 𝑦 𝑢

 𝑏𝑙 is how much of each harmonic

is required to construct the periodic signal 𝑦(𝑢)

VECTOR SPACE OF PERIODIC SIGNALS

HARMONICS

 𝑙 = ±1 ⇒ fundamental component (first harmonic)

 Frequency 𝜕0, period 𝑈 = 2𝜌/𝜕0

 𝑙 = ±2 ⇒ second harmonic

 Frequency 𝜕2 = 2𝜕0, period 𝑈2 = 𝑈/2 (half period)

 …  𝑙 = ±𝑂 ⇒ Nth harmonic

 Frequency 𝜕𝑂 = 𝑂𝜕0, period 𝑈𝑂 = 𝑈/𝑂 (1/N period)

 𝑙 = 0 ⇒ 𝑏0 =

1 𝑈 ∫ 𝑈 𝑦 𝑢 𝑒𝑢, DC, constant component, average

HOW TO FIND FS REPRESENTATION

 Will use important examples to demonstrate

common techniques

 Sinusoidal signals – Euler’s relationship  Direct FS integral evaluation  FS properties table and transform pairs

coefficients by inspection

SINUSOIDAL SIGNAL

1 𝑢 < 𝑈

𝑈

PERIODIC RECTANGLE WAVE

 Important signal/function in

DSP and communication

 sinc 𝑦 =

normalized

 sinc 𝑦 =

unnormalized

 Modulated sine function

 Amplitude follows 1/x  Must use L’Hopital’s rule to get

x=0 time

SINC FUNCTION

 Consider different “duty cycle” for

the rectangle wave

 𝑈 = 4𝑈

 𝑈 = 8𝑈

 𝑈 = 16𝑈

 Note all plots are still a sinc

shape

 Difference is how the sync is sampled  Longer in time (larger T) smaller

spacing in frequency  more samples between zero crossings

RECTANGLE WAVE COEFFICIENTS

 Special case of rectangle wave

with 𝑈 = 4𝑈

1

 One sample between zero-crossing

 𝑏𝑙 = ቐ

1/2 𝑙 = 0

sin(𝑙𝜌/2) 𝑙𝜌

𝑓𝑚𝑡𝑓

SQUARE WAVE

 𝑦 𝑢 = σ𝑙=−∞

∞

𝜀(𝑢 − 𝑙𝑈)

 Using FS integral

PERIODIC IMPULSE TRAIN

PROPERTIES OF CTFS

 Since these are very similar between CT and DT,

will save until after DT

 Note: As for LT and Z Transform, properties are