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:
Let
$\mathcal{F}(\xb7)$ denote the transformation from
$f(t)$ to the Fourier coefficients
$$\mathcal{F}(f(t))=\forall n, n\in \mathbb{Z}\colon {c}_{n}$$$\mathcal{F}(\xb7)$ maps complex valued functions to sequences of
complex numbers .
Linearity
$\mathcal{F}(\xb7)$ is a
linear transformation .
If
$\mathcal{F}(f(t))={c}_{n}$ and
$\mathcal{F}(g(t))={d}_{n}$ .
Then
$$\forall \alpha , \alpha \in \mathbb{C}\colon \mathcal{F}(\alpha f(t))=\alpha {c}_{n}$$ and
$$\mathcal{F}(f(t)+g(t))={c}_{n}+{d}_{n}$$
Easy. Just linearity of integral.
$\mathcal{F}(f(t)+g(t))=\forall n, n\in \mathbb{Z}\colon \int_{0}^{T} (f(t)+g(t))e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega}_{0}nt)}\,d t+\frac{1}{T}\int_{0}^{T} g(t)e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon {c}_{n}+{d}_{n}={c}_{n}+{d}_{n}$
$\mathcal{F}(f(t-{t}_{0}))=e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}$ if
${c}_{n}=\left|{c}_{n}\right|e^{i\angle ({c}_{n})}$ ,
then
$$\left|e^{-(i{\omega}_{0}n{t}_{0})}{c}_{n}\right|=\left|e^{-(i{\omega}_{0}n{t}_{0})}\right|\left|{c}_{n}\right|=\left|{c}_{n}\right|$$$$\angle (e^{-(i{\omega}_{0}{t}_{0}n)})=\angle ({c}_{n})-{\omega}_{0}{t}_{0}n$$
$\mathcal{F}(f(t-{t}_{0}))=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{0}^{T} f(t-{t}_{0})e^{-(i{\omega}_{0}nt)}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(t-{t}_{0})e^{-(i{\omega}_{0}n(t-{t}_{0}))}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon \frac{1}{T}\int_{-{t}_{0}}^{T-{t}_{0}} f(\stackrel{~}{t}())e^{-(i{\omega}_{0}n\stackrel{~}{t})}e^{-(i{\omega}_{0}n{t}_{0})}\,d t=\forall n, n\in \mathbb{Z}\colon e^{-(i{\omega}_{0}n\stackrel{~}{t})}{c}_{n}$
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\to $∞ .If
$\mathcal{F}(f(t))={c}_{n}$ and
$\left|{c}_{n}\right|$ has the form
$\frac{1}{n^{k}}$ ,
then
$\mathcal{F}(\frac{d^{m}f(t)}{dt^{m}})=(in{\omega}_{0})^{m}{c}_{n}$ and has the form
$\frac{n^{m}}{n^{k}}$ .So for the
${m}^{\mathrm{th}}$ derivative to have finite energy, we need
$$\sum \left|\frac{n^{m}}{n^{k}}\right|^{2}$$∞ thus
$\frac{n^{m}}{n^{k}}$ decays
faster than
$\frac{1}{n}$ which implies that
$$2k-2m> 1$$ or
$$k> \frac{2m+1}{2}$$ Thus the decay rate of the Fourier series dictates
smoothness.
Fourier differentiation demonstration
Integration in the fourier domain
If
$\mathcal{F}(f(t))={c}_{n}$
then
$\mathcal{F}(\int_{()} \,d \tau )$∞tfτ1ω0ncn
If
${c}_{0}\neq 0$ , 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
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}=\sum_{k=()} $∞∞ckdn-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
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
$ax\left(t\right)+by\left(t\right)$
$aX\left(f\right)+bY\left(f\right)$
Time Shifting
$x(t-\tau )$
$X\left(f\right){e}^{-j2\pi f\tau /T}$
Time Modulation
$x\left(t\right){e}^{j2\pi f\tau /T}$
$X(f-k)$
Multiplication
$x\left(t\right)y\left(t\right)$
$X\left(f\right)*Y\left(f\right)$
Continuous Convolution
$x\left(t\right)*y\left(t\right)$
$X\left(f\right)Y\left(f\right)$
Questions & Answers
What is price elasticity of demand and its degrees. also explain factors determing price elasticity of demand?
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.
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.
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.
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
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....
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
Got questions? Join the online conversation and get instant answers!