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An introduction to the general properties of the Fourier series

Introduction

In this module we will discuss the basic properties of the Continuous-Time Fourier Series. We will begin by refreshing your memory of our basic Fourier series equations:

f t n c n ω 0 n t
c n 1 T t 0 T f t ω 0 n t
Let · denote the transformation from f t to the Fourier coefficients f t n n c n · maps complex valued functions to sequences of complex numbers .

Linearity

· is a linear transformation .

If f t c n and g t d n . Then α α α f t α c n and f t g t c n d n

Easy. Just linearity of integral.

f t g t n n t 0 T f t g t ω 0 n t n n 1 T t 0 T f t ω 0 n t 1 T t 0 T g t ω 0 n t n n c n d n c n d n

Shifting

Shifting in time equals a phase shift of Fourier coefficients

f t t 0 ω 0 n t 0 c n if c n c n c n , then ω 0 n t 0 c n ω 0 n t 0 c n c n ω 0 t 0 n c n ω 0 t 0 n

f t t 0 n n 1 T t 0 T f t t 0 ω 0 n t n n 1 T t t 0 T t 0 f t t 0 ω 0 n t t 0 ω 0 n t 0 n n 1 T t t 0 T t 0 f t ~ ω 0 n t ~ ω 0 n t 0 n n ω 0 n t ~ c n

Parseval's relation

t 0 T f t 2 T n c n 2
Parseval's relation tells us that the energy of a signal is equal to the energy of its Fourier transform.
Parseval tells us that the Fourier series maps L 0 T 2 to l 2 .

For f t to have "finite energy," what do the c n do as n ?

c n 2 for f t to have finite energy.

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If n n 0 c n 1 n , is f L 0 T 2 ?

Yes, because c n 2 1 n 2 , which is summable.

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Now, if n n 0 c n 1 n , is f L 0 T 2 ?

No, because c n 2 1 n , which is not summable.

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The rate of decay of the Fourier series determines if f t has finite energy .

Parsevals theorem demonstration

ParsevalsDemo
Interact (when online) with a Mathematica CDF demonstrating Parsevals Theorem. To download, right click and save file as .cdf.

Symmetry properties

Even signals

    Even signals

  • f ( t ) = f ( - t )
  • c n = c - n
  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t

Odd signals

    Odd signals

  • f ( t ) = -f ( -t )
  • c n = c - n *
  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t - 1 T T 2 T f ( - t ) exp ( ı ω 0 n t ) d t
  • = - 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t - exp ( - ı ω 0 n t ) d t
  • = - 1 T 0 T f ( t ) 2 ı sin ( ω 0 n t ) d t

Real signals

    Real signals

  • f ( t ) = f * ( t )
  • c n = c - n *
  • c n = 1 T 0 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T 2 f ( - t ) exp ( - ı ω 0 n t ) d t + 1 T T 2 T f ( - t ) exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) exp ( ı ω 0 n t ) d t + exp ( - ı ω 0 n t ) d t
  • = 1 T 0 T f ( t ) 2 cos ( ω 0 n t ) d t

Differentiation in fourier domain

f t c n t f t n ω 0 c n

Since

f t n c n ω 0 n t
then
t f t n c n t ω 0 n t n c n ω 0 n ω 0 n t
A differentiator attenuates the low frequencies in f t and accentuates the high frequencies. It removes general trends and accentuates areas of sharpvariation.
A common way to mathematically measure the smoothness of a function f t is to see how many derivatives are finite energy.
This is done by looking at the Fourier coefficients of thesignal, specifically how fast they decay as n .If f t c n and c n has the form 1 n k , then t m f t n ω 0 m c n and has the form n m n k .So for the m th derivative to have finite energy, we need n n m n k 2 thus n m n k decays faster than 1 n which implies that 2 k 2 m 1 or k 2 m 1 2 Thus the decay rate of the Fourier series dictates smoothness.

Fourier differentiation demonstration

FourierDiffDemo
Interact (when online) with a Mathematica CDF demonstrating Differentiation in the Fourier Domain. To download, right click and save file as .cdf.

Integration in the fourier domain

If

f t c n
then
τ t f τ 1 ω 0 n c n
If c 0 0 , this expression doesn't make sense.

Integration accentuates low frequencies and attenuates high frequencies. Integrators bring out the general trends in signals and suppress short term variation (which is noise in many cases). Integrators are much nicer than differentiators.

Fourier integration demonstration

fourierIntDemo
Interact (when online) with a Mathematica CDF demonstrating Integration in the Fourier Domain. To download, right click and save file as .cdf.

Signal multiplication and convolution

Given a signal f t with Fourier coefficients c n and a signal g t with Fourier coefficients d n , we can define a new signal, y t , where y t f t g t . We find that the Fourier Series representation of y t , e n , is such that e n k c k d n - k . This is to say that signal multiplication in the time domainis equivalent to signal convolution in the frequency domain, and vice-versa: signal multiplication in the frequency domain is equivalent to signal convolution in the time domain.The proof of this is as follows

e n 1 T t 0 T f t g t ω 0 n t 1 T t 0 T k c k ω 0 k t g t ω 0 n t k c k 1 T t 0 T g t ω 0 n k t k c k d n - k
for more details, see the section on Signal convolution and the CTFS

Conclusion

Like other Fourier transforms, the CTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.

Properties of the ctfs
Property Signal CTFS
Linearity a x ( t ) + b y ( t ) a X ( f ) + b Y ( f )
Time Shifting x ( t - τ ) X ( f ) e - j 2 π f τ / T
Time Modulation x ( t ) e j 2 π f τ / T X ( f - k )
Multiplication x ( t ) y ( t ) X ( f ) * Y ( f )
Continuous Convolution x ( t ) * y ( t ) X ( f ) Y ( f )

Questions & Answers

What is price elasticity of demand and its degrees. also explain factors determing price elasticity of demand?
Yutansh Reply
Price elasticity of demand (PED) is use to measure the degree of responsiveness of Quantity demanded for a given change on price of the good itself, certis paribus. The formula for PED = percentage change in quantity demanded/ percentage change in price of good A
GOH
its is necessarily negative due to the inverse relationship between price and Quantity demanded. since PED carries a negative sign most of the time, we will usually the absolute value of PED by dropping the negative sign.
GOH
PED > 1 means that the demand of the good is price elasticity and for a given increase in price there will be a more then proportionate decrease in quantity demanded.
GOH
PED < 1 means that the demand of the good is price inelasticity and for a given increase in price there will be a less then proportionate decrease in quantity demanded.
GOH
The factors that affects PES are: Avaliablilty of close substitutes, proportion of income spent on the good, Degree of necessity, Addiction and Time.
GOH
Calculate price elasticity of demand and comment on the shape of the demand curve of a good ,when its price rises by 20 percentage, quantity demanded falls from 150 units to 120 units.
Helen Reply
5 %fall in price of good x leads to a 10 % rise in its quantity demanded. A 20 % rise in price of good y leads to do a 10 % fall in its quantity demanded. calculate price elasticity of demand of good x and good y. Out of the two goods which one is more elastic.
Helen
what is labor
Grace Reply
labor is any physical or mental effort that helps in the production of goods and services
Kwabena
what is profit maximizing level of out put for above hypothetical firm TC = Q3 - 21Q2 + 600 + 1800 P = 600 MC = 3Q2 - 42Q + 600
Sosna Reply
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
Sosna
sorry it the mistake answer it is question
Sosna
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
Sosna
The formula for calculation income elasticity of demand is the percent change in quantity demanded divided by the percent change in income.
Sosna
what is labor productivity
Lizzy Reply
if the demand function is q=25-4p+p² 1.find elasticity of demand at the point p=5?
Puja Reply
what are some of the difference between monopoly and perfect competition market
Obeng Reply
n a perfectly competitive market, price equals marginal cost and firms earn an economic profit of zero. In a monopoly, the price is set above marginal cost and the firm earns a positive economic profit. Perfect competition produces an equilibrium in which the price and quantity of a good is economic
Naima
what are some characteristics of monopoly market
Obeng Reply
explicit cost is seen as a total experiences in the business or the salary (wages) that a firm pay to employee.
Idagu Reply
what is price elasticity
Fosua
...
krishna
it is the degree of responsiveness to a percentage change in the price of the commodity
Obeng
economics is known to be the field
John Reply
what is monopoly
Peter Reply
what is taxation
Peter
is the compulsory transfer of wealth from the private sector to the public sector
Jonna
why do monopoly make excess profit in both long run and short run
Adeola Reply
because monopoly have no competitor on the market and they are price makers,therefore,they can easily increase the princes and produce small quantity of goods but still consumers will still buy....
Kennedy
how to identify a perfect market graph
Adeola Reply
what is the investment
jimmy
investment is a money u used to the business
Mohamed
investment is the purchase of good that are not consumed today but are used in the future to create wealth.
Amina
investment is the good that are not consumed
Fosua
What is supply
Fosua
 Supply represents how much the market can offer.
Yusif
it is the quantity of commodity producers produces at the market
Obeng
what is the effect of scarce resources on producers
Phindu Reply
explain how government taxes and government producer subsidies affect supply
Chanda
what is economic
Charles Reply
what are the type of economic
Charles
macroeconomics,microeconomics,positive economics and negative economics
Gladys
what are the factors of production
Gladys
process of production
Mutia
Basically factors of production are four (4) namely: 1. Entrepreneur 2. Capital 3. Labour and; 4. Land but there has been a new argument to include an addition one to the the numbers to 5 which is "Technology"
Elisha
what is land as a factor of production
Gladys
what is Economic
Abu
economics is how individuals bussiness and governments make the best decisions to get what they want and how these choices interact in the market
Nandisha
Economics as a social science, which studies human behaviour as a relationship between ends and scarce means, which have alternative uses.
Yhaar
Economics is a science which study human behaviour as a relationship between ends and scarce means
John
Economics is a social sciences which studies human behavior as a relationship between ends and scarce mean, which have alternative uses.....
Pintu
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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