# 9.1 Discrete time fourier transform (dtft)

 Page 1 / 1
Details the discrete-time fourier transform.

## Introduction

In this module, we will derive an expansion for arbitrary discrete-time functions, and in doing so, derive the Discrete Time Fourier Transform (DTFT).

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems , calculating the output of an LTI system $ℋ$ given $e^{i\omega n}$ as an input amounts to simple multiplication, where ${\omega }_{0}=\frac{2\pi k}{N}$ , and where $H(k)\in \mathbb{C}$ is the eigenvalue corresponding to k. As shown in the figure, a simple exponential input would yield the output

$y(n)=H(k)e^{i\omega n}$

Using this and the fact that $ℋ$ is linear, calculating $y(n)$ for combinations of complex exponentials is also straightforward.

${c}_{1}e^{i{\omega }_{1}n}+{c}_{2}e^{i{\omega }_{2}n}\to {c}_{1}H({k}_{1})e^{i{\omega }_{1}n}+{c}_{2}H({k}_{2})e^{i{\omega }_{1}n}$ $\sum {c}_{l}e^{i{\omega }_{l}n}\to \sum {c}_{l}H({k}_{l})e^{i{\omega }_{l}n}$

The action of $H$ on an input such as those in the two equations above is easy to explain. $ℋ$ independently scales each exponential component $e^{i{\omega }_{l}n}$ by a different complex number $H({k}_{l})\in \mathbb{C}$ . As such, if we can write a function $y(n)$ as a combination of complex exponentials it allows us to easily calculate the output of a system.

Now, we will look to use the power of complex exponentials to see how we may represent arbitrary signals in terms of a set of simpler functions bysuperposition of a number of complex exponentials. Below we will present the Discrete-Time Fourier Transform (DTFT). Because the DTFT deals with nonperiodic signals, we must find away to include all real frequencies in the general equations.For the DTFT we simply utilize summation over all real numbers rather thansummation over integers in order to express the aperiodic signals.

## Dtft synthesis

It can be demonstrated that an arbitrary Discrete Time-periodic function $f(n)$ can be written as a linear combination of harmonic complex sinusoids

$f(n)=\sum_{k=0}^{N-1} {c}_{k}e^{i{\omega }_{0}()kn}$
where ${\omega }_{0}=\frac{2\pi }{N}$ is the fundamental frequency. For almost all $f(n)$ of practical interest, there exists ${c}_{n}$ to make [link] true. If $f(n)$ is finite energy ( $f(n)\in L(0, N)^{2}$ ), then the equality in [link] holds in the sense of energy convergence; with discrete-time signals, there are no concerns for divergence as there are with continuous-time signals.

The ${c}_{n}$ - called the Fourier coefficients - tell us "how much" of the sinusoid $e^{j{\omega }_{0}kn}$ is in $f(n)$ . The formula shows $f(n)$ as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set ofcomplex exponentials $\{\forall k, k\in \mathbb{Z}\colon e^{j{\omega }_{0}kn}\}$ form a basis for the space of N-periodic discrete time functions.

## Equations

Now, in order to take this useful tool and apply it to arbitrary non-periodic signals, we will have to delve deeper into the use of the superposition principle. Let ${s}_{T}\left(t\right)$ be a periodic signal having period $T$ . We want to consider what happens to this signal's spectrum as the period goes to infinity. We denote the spectrum for any assumed value of the period by ${c}_{n}\left(T\right)$ . We calculate the spectrum according to the Fourier formula for a periodic signal, known as the Fourier Series (for more on this derivation, see the section on Fourier Series .)

${c}_{n}=\frac{1}{T}{\int }_{0}^{T}\phantom{\rule{-0.166667em}{0ex}}s\left(t\right)exp\left(-ı{\omega }_{0}t\right)\phantom{\rule{0.166667em}{0ex}}dt$
where ${\omega }_{0}=\frac{2\pi }{T}$ and where we have used a symmetric placement of the integration interval about the origin for subsequent derivational convenience. We vary the frequency index $n$ proportionally as we increase the period. Define making the corresponding Fourier Series
${s}_{T}\left(t\right)=\sum _{-\infty }^{\infty }\phantom{\rule{-0.166667em}{0ex}}f\left(t\right)exp\left(ı{\omega }_{0}t\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{T}\right)$
As the period increases, the spectral lines become closer together, becoming a continuum. Therefore,
$\underset{T\to \infty }{lim}{s}_{T}\left(t\right)\equiv s\left(t\right)=\underset{-\infty }{\overset{\infty }{\int }}\phantom{\rule{-0.166667em}{0ex}}S\left(f\right)exp\left(ı{\omega }_{0}t\right)\phantom{\rule{0.166667em}{0ex}}df$
with
$S\left(f\right)=\underset{-\infty }{\overset{\infty }{\int }}\phantom{\rule{-0.166667em}{0ex}}s\left(t\right)exp\left(-ı{\omega }_{0}t\right)\phantom{\rule{0.166667em}{0ex}}dt$

## Discrete-time fourier transform

$ℱ(\omega )=\sum_{n=()}$ f n ω n

## Inverse dtft

$f(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi } ℱ(\omega )e^{i\omega n}\,d \omega$

It is not uncommon to see the above formula written slightly different. One of the most common differences is the way that the exponential is written. The above equations use the radial frequency variable $\omega$ in the exponential, where $\omega =2\pi f$ , but it is also common to include the more explicit expression, $i2\pi ft$ , in the exponential. Sometimes DTFT notation is expressed as $F(e^{i\omega })$ , to make it clear that it is not a CTFT (which is denoted as $F(\Omega )$ ). Click here for an overview of the notation used in Connexion's DSP modules.

## Dtft summary

Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The discrete time Fourier transform synthesis formula expresses a discrete time, aperiodic function as the infinite sum of continuous frequency complex exponentials.

$ℱ(\omega )=\sum_{n=()}$ f n ω n
The discrete time Fourier transform analysis formula takes the same discrete time domain signal and represents the signal in the continuous frequency domain.
$f(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi } ℱ(\omega )e^{i\omega n}\,d \omega$

#### Questions & Answers

What is deflation
Enumerate emotional intelligence to a manager
What about Sydney Alexander's Absorption approach in international trade?
I need help in inflation graphs
Select inflation type, Demand pull, cost pull or anticipation 1- Select the set of data you intend on graphing i.e inflation rate of 2017, location (particular country) 2 - Select the type of measurement tool that best allows you to input the inflation data, Consumer price index is the most accurate
Jama
this is to make sure you have all the correct information, Also use should know 1- Cost pull is Aggregate Demand and Aggregate Supply AD - AS graphed 2- Demans pull is Aggregate Supply and Aggregate Demand AS - AD graphed
Jama
what is production
what is a monopolistic competition?
Enumerate emotional intelligence to a manager
Chinonso
Enumerate emotional intelligence to a manager
Chinonso
who is barter
exchange goods each other
Seven
what is economic
Bah
is the use of scares resources to satisfy our unlimited needs and wants
Desiderius
how many kinds of utility functions?
What is partnership?
Jackson
the legal association of two or more people as co-owners of a business for profit.
harmony
Would you expect the kinked demand curve to be more extreme (like a right angle) or less extreme (like a normal demand curve) if each firm in the cartel produces a near-identical product like OPEC and petroleum? What if each firm produces a somewhat different product?
no
what is supply
what is opportunity cost
Mizta
The opportunity gained interms of opportunity lost is known as opportunity cost Or The second best alternative use of resources
Mir
forgone alternative: like forgoing Something our of two to buy one
Tam-Waribo
what is macro economic s
macroeconomics is the study of economic as a whole level.
Gafar
meaning of positive science
positive science it is focused on facts and cause and effect and behavioural relationship and include developmental testing in economic theoreis.
Gafar
what is inflation
inflation is the general price increase of goods and services in an economy.
tesfie
Inflation is the persistent rise in the general price level
T-Max
inflation is characterized by increase in the general price of goods and services. when there is too much money in circulation. increase in demand of goods pursuing fewer goods. when purchasing power of money decreases .
Ejikeme
inflation is the persistent rise general price level
Habeeb
inflation is the persistent increase in price
Machall
hi
Rafiu
yes
boston
hi
Ayaan
how are you
Ayaan
increase in the general level of price...
what is deflation
Sele
is the gradual decrease of currency exchange in a country.
Gafar
why ecnomics important ? give answer plz
Saifullah
Because is a field of science study that reflects on our day to day activities with human behavior.
ANSU
why economic is a science
Imoro
Economics is referred to as a social science not a pure science. It's regarded as a social science because it makes use of the scientific method to solve problems. The scientific method refers to observation, asking questions, forming hypothesis, experimentation etc
Nkechi
Economics is a social science because it study human behavior how he relates with his daily activities with the available limited resources to satisfy his wants.
Ojo
what are the factors affecting the demand
Mohammed
yeah it uses the scientific method to study human behaviour.
Nkechi
inflation referes to the persistant increase in the general price of goods and services over a given period of time say a year.
Abdul
factors affacting Demand of good and services are 1.price of a commodity in question 2.price of related commodity 3.Income of a consumer 4.Population 5.tast and prefereance 6.Season or weather condition
Abdul
what is difference between perfect and non perfect market.
Saheed
what the difference between Trade off and Opportunity Cost?
Elzevery
Golda
In trade off, you increase the amount of something by decreasing the amount of something else. For example, you use 2 hours to study and 2 hours for leisure. if you increase study hour by 1 more hour, i.e 3 hours, leisure time will decrease by 1 hour, i.e 1 hour.
harmony
In all, you would have traded off 1 hour of leisure time for 3 hours of study time. But in opportunity cost, you let something go in order to obtain something else entirely.
harmony
thanks for your idea
Elzevery
please i want help on thid question given P=\$10 And TC=120+4Q2 1.find the profit maximizing level of price and quantity. 2.what will be the total profit?
Shemels
please how is substitutional effect affecting demand
Acha
if a price of a particular commodity is high people demand less ,they rather go for less one
FIDELIS
what is account
Wasif
account is an arrangement between a customer and a bank that allows the customer to play in and take out money (bank account)
Johnny
pay, not play sorry
Johnny
four effects of inflation in an economy
Acha
oil
TsendeTheRipper
decreases living standards .. decreases purchasing power.. Decreases internation competition . increases the cost of borrowing
MansoorAfghan
Inflation refers to persistent increase in the general prices of commodities.
Aluko
he asked for effects
MansoorAfghan
Low standard of living, prices of commodities will be high, devaluation of currency in the economy
Aluko
Debtors gain while creditors loses
Ojo
Hi every one
Ani
Hello
Ansah
hi
kelvin
hi
Seven
Hii
raushan
how are you all
Seven
Fin and you
raushan
me also
Seven
Whare do you live
raushan
general tendency of rising price and continuous... .
Auqib
any one among you is going to aaper in net exam day after tomorrow
Auqib
Auqib
pakistan
Seven
Oo Pakistan I am India
raushan
really from Pakistan
Auqib
ok
Seven
You noo hindi
raushan
no
Seven
hi
ken
what is deference between the producer and firm?
ISMAILI
A producer, going by its name, only makes tangible goods and services from the use of raw materials or from the ground up. Meanwhile, a firm can be a wholesale corporation can depends directly on the finished goods and services of the producer in order to resell to the market.
harmony
Okay! I got a point.. so can you give an examples Harmony?
ISMAILI
An example for the producer would be Microsoft Corporation that makes Windows Operating System for computer-making firms like Acer, Dell, Hp, etc.
harmony
Calculate the elasticity coefficient when price decrease from 60 to 50 and 40 to 30 interpret the result
Acha
Thanks for your ideas bro..
ISMAILI
My pleasure, Shabani.
harmony
hi
Habib
hello rum I can see some great economists here
olasupo
And how does this thing even work
MansoorAfghan
does everyone on the app get the questioms
MansoorAfghan
ns
MansoorAfghan
mec
Abdul
different between demand and quantity demand
No difference
MansoorAfghan
demand is the overall demand for it
MansoorAfghan
actually theres no difference
MansoorAfghan
quantity demanded is used in Equilibrium of d and s
MansoorAfghan
for evrything else u use deman
MansoorAfghan
the difference of it is that when demand simply denotes the willingness and a person's ability to purchase. And as against quantity demand represent the amount of an economic good or services desire by a consumer at a fixed price .☺
Gafar
how to calculate inflation
Explain the factors that have led to high quantity demanded
price of the product increase of price substitute product as people shift to cheap one
Black
Give examples of non-banking institution.
Aluko
microfinance bank is one good example of non-banking institution
Odeniyi
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Got questions? Join the online conversation and get instant answers!