Week 2, Lecture A: CT Fourier Series (Trig Form) · 1=2 1=2 0 1 x(t) 9% You can decrease (and even...

6.003Signal Processing

Week 2, Lecture A:

CT Fourier Series(Trig Form)

Adam [email protected]

Transforms

Signals are functions that convey information. However, thestraightforward way of representing a signal may not exposeimportant properties of the signal.

It is often useful to have multiple different ways of lookingat a signal.

6.003 Signal Processing Week 2 Lecture A (slide 2) 12 Feb 2019

Transforms

For example, consider the following two representations ofa speech signal:


Transforms

We call such an alternative view of a signal a transform.

In 6.003, we will primarily focus on a family of transformsreferred to as Fourier transforms, where signals arerepresented as sums of harmonically-related sinusoids, forexample:

f(t) = c0 +

∞∑k=1

(ck cos(kωot) + dk sin(kωot)

)

PSet 1 contained an example of this. Given a bunch ofcoefficients associated with harmonically-related sinusoids,find the waveform that results from combining themtogether.

Today: how can we go in the other direction?Under what conditions can f(t) be expressed in this form?How can we compute ck and dk?


Preliminaries: Jargon

A periodic signal is one that repeats after some amount oftime T , such that, for all t,

x(t) = x(t+ T )

A function that is periodic in T is also periodic in2T, 3T, 4T, . . .. For a given signal, the smallest value T forwhich the above holds is called the signal’s fundamentalperiod.

The associated frequency, ω0 = 2πT, is the signal’s

fundamental frequency.

Frequencies are harmonically related if they are integermultiples of some fundamental frequency.


Preliminaries: Sinusoids

A cos(ωt) +B sin(ωt) =(√

A2 +B2)cos

(ωt+ tan−1

(−b

a

))



Where k is a positive integer:∫ t0+T

t0

sin

(2πk

Tt

)dt = 0

∫ t0+T

t0

cos

(2πk

Tt

)dt = 0



Where k and m are positive integers:∫ t0+T

t0

sin

(2πk

Tt

)cos

(2πm

Tt

)dt = 0



Where k and m are positive integers:

∫ t0+T

t0

cos

(2πk

Tt

)cos

(2πm

Tt

)dt =

{T/2 if k = m,

0 otherwise



Where k and m are positive integers:

∫ t0+T

t0

sin

(2πk

Tt

)sin

(2πm

Tt

)dt =

{T/2 if k = m,

0 otherwise


Preliminaries

Fourier series are sums of harmonically related sinusoids:

f(t) = c0 +

∞∑k=1

(ck cos(kωot) + dk sin(kωot)

)where ωo represents the “fundamental frequency.”

Basis functions:

2πωo

t

cos0t

2πωo

t

sin0t

2πωo

t

cosωot

2πωo

tsinωot

2πωo

t

cos2ω

ot

...2πωo

t

sin2ωot

...


Fourier Representations of Signals

What signals can be represented by sums of harmoniccomponents?

ωωo 2ωo 3ωo 4ωo 5ωo 6ωo

T= 2πωo

t

T= 2πωo

t

Only periodic signals: all harmonics of ωo are periodic inT = 2π/ωo.


Fourier Representations of Signals

Assume f(t) is periodic in T and is composed of a weightedsum of harmonics of w0 = 2π/T :

f(t) = c0 +

∞∑k=1

(ck cos

(2πk

Tt

)+ dk sin

(2πk

Tt

))How do we find the weights?


c0: The “DC” Term

The constant c0 is referred to as the “DC” term: itrepresents the constant offset of a signal (if any). Statedanother way, it represents the average value of a signal overone period.

c0 =1

T

∫ t0+T

t0

f(t)dt


Computing the other ck terms

We are assuming that our signal is comprised ofharmonically-related sines and cosines. Here is an examplesignal:

f(t) = c1 cos(ω0t) + d1 sin(ω0t) + c3 cos(3ω0t) + d5 sin(5ω0t)

Let’s imagine we wanted to determine the coefficient c3,associated with cos(3ω0t).

Our strategy will be to multiply f(t) by cos(3ω0t) andintegrate the result over one period.

Check Yourself: How does this help us find c3?


Computing the other terms

In general, we can compute the coefficients:

ck =2

T

∫ t0+T

t0

f(t) cos

(2πkt

T

)dt

We can use similar logic to find the coefficients associatedwith the sine waves:

dk =2

T

∫ t0+T

t0

f(t) sin

(2πkt

T

)dt


Fourier

Complicated waveforms represented as sums ofharmonically-related sinusoids. Two “views”:

Frequency

Time

Magnitude


Moving Between Representations

Synthesis:

f(t) = c0 +

∞∑k=1

(ck cos

(2πk

Tt

)+ dk sin

(2πk

Tt

))

Analysis:

c0 =1

T

∫ t0+T

t0

f(t)dt

ck =2

T

∫ t0+T

t0

f(t) cos

(2πkt

T

)dt

dk =2

T

∫ t0+T

t0

f(t) sin

(2πkt

T

)dt


Check Yourself!

What are the Fourier series coefficients associated with thefollowing signal?

f(t) = 0.8 sin(6πt)− 0.3 cos(6πt) + 0.75 cos(12πt)


Check Yourself!

What are the Fourier series coefficients associated with thefollowing signal?

f(t) = cos(2πt− π/4)


Check Yourself!

Let ck and dk be the Fourier series coefficients associatedwith the following signal:

t

1/2

−1/2

0 1

x(t)

Which of the following statements are true?

• ck = 0 for all k

• dk = 0 if k is even

• |dk| decreases with k2

• there are an infinite number of non-zero dk


Gibbs Phenomenon

Partial sums of Fourier series of discontinuous functions“ring” near discontinuities: Gibbs’ phenomenon.

t

1/2

−1/2

0 1

x(t) 9%

You can decrease (and even eliminate the ringing) bydecreasing the magnitudes of the Fourier coefficients athigher frequencies.


Summary

Today, we learned about the CT Fourier series, which is away of representing signals as sums of sinusoidalcomponents.

Recitation 2A: Practice with finding Fourier series

Lecture 2B: CT Fourier Series, Complex Exponential Form


Week 2, Lecture A: CT Fourier Series (Trig Form) · 1=2 1=2 0 1 x(t) 9% You can decrease (and even...

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