This module describes the circular convolution algorithm and an alternative algorithm

Introduction

This module relates circular convolution of periodic signals in one domain to multiplication in the other domain.

You should be familiar with
Discrete-Time Convolution , which tells us
that given two discrete-time signals
$x(n)$ , the system's input, and
$h(n)$ , the system's response, we define the output of the
system as

$y(n)=(x(n), h(n))=\sum_{k=()} $∞∞xkhnk

When we are given two DFTs (finite-length sequences usually oflength
$N$ ), we cannot just
multiply them together as we do in the above convolutionformula, often referred to as
linear convolution .
Because the DFTs are periodic, they have nonzero values for
$n\ge N$ and thus the multiplication of these two DFTs will
be nonzero for
$n\ge N$ . We need to define a new type of convolution
operation that will result in our convolved signal being zerooutside of the range
$n=\{0, 1, \dots , N-1\}$ . This idea led to the development of
circular
convolution , also called cyclic or periodic convolution.

Signal circular convolution

Given a signal
$f(n)$ with Fourier coefficients
${c}_{k}$ and a signal
$g(n)$ with Fourier coefficients
${d}_{k}$ ,
we can define a new signal,
$v(n)$ ,
where
$v(n)=\u229b(f(n), g(n))$ We find that the
Fourier
Series representation of
$v(n)$ ,
${a}_{k}$ ,
is such that
${a}_{k}={c}_{k}{d}_{k}$ .
$\u229b(f(n), g(n))$ is the
circular convolution of two periodic signals and is equivalent to the convolution
over one interval,
i.e.$\u229b(f(n), g(n))=\sum_{n=0}^{N} \sum_{\eta =0}^{N} f(\eta )g(n-\eta )$ .

Circular convolution in the time domain is equivalent to
multiplication of the Fourier coefficients.

where we can reduce the second summation found in the above
equation into
$h({\left(\right(n-m\left)\right)()}_{N})=\frac{1}{N}\sum_{k=0}^{N-1} H(k)e^{(j\frac{2\pi}{N}k(n-m))}$$y(n)=\sum_{m=0}^{N-1} f(m)h({\left(\right(n-m\left)\right)()}_{N})$ which equals circular convolution! When we have
$0\le n\le N-1$ in the above, then we get:

$y(n)\equiv \u229b(f(n), h(n))$

The notation
$\u229b$ represents cyclic convolution "mod N".

Alternative convolution formula

Alternative circular convolution algorithm

Step 1: Calculate the DFT of
$f(n)$ which yields
$F(k)$ and calculate the DFT of
$h(n)$ which yields
$H(k)$ .

Step 2: Pointwise multiply
$Y(k)=F(k)H(k)$

Step 3: Inverse DFT
$Y(k)$ which yields
$y(n)$

Seems like a roundabout way of doing things,
but it turns out that there are
extremely fast ways to calculate the
DFT of a sequence.

To circularily convolve
$2$$N$ -point sequences:
$y(n)=\sum_{m=0}^{N-1} f(m)h({\left(\right(n-m\left)\right)()}_{N})$ For each
$n$ :
$N$ multiples,
$N-1$ additions

We can picture
periodic sequences as having discrete
points on a circle as the domain

Shifting by
$m$ ,
$f(n+m)$ , corresponds to rotating the cylinder
$m$ notches ACW (counter
clockwise). For
$m=-2$ , we get a shift equal to that in the following
illustration:

To cyclic shift we follow these steps:

1) Write
$f(n)$ on a cylinder, ACW

2) To cyclic shift by
$m$ , spin
cylinder m spots ACW
$$\to (f(n), f({((n+m))}_{N}))$$

Notes on circular shifting

$f({((n+N))}_{N})=f(n)$ Spinning
$N$ spots is the same
as spinning all the way around, or not spinning at all.

$f({((n+N))}_{N})=f({((n-(N-m)))}_{N})$ Shifting ACW
$m$ is equivalent to
shifting CW
$N-m$

$f({((-n))}_{N})$ The above expression, simply writes the values of
$f(n)$ clockwise.

Convolve (n = 4)

$h({\left((\right(, -, m, \left), \right))}_{N})$

Multiply
$f(m)$ and
$\mathrm{sum}$ to yield:
$y(0)=3$

$h({\left((\right(, 1, -, m, \left), \right))}_{N})$

Multiply
$f(m)$ and
$\mathrm{sum}$ to yield:
$y(1)=5$

$h({\left((\right(, 2, -, m, \left), \right))}_{N})$

Multiply
$f(m)$ and
$\mathrm{sum}$ to yield:
$y(2)=3$

$h({\left((\right(, 3, -, m, \left), \right))}_{N})$

Multiply
$f(m)$ and
$\mathrm{sum}$ to yield:
$y(3)=1$

For this signal
${c}_{k}=\begin{cases}\frac{1}{N} & \text{if $k=0$}\\ \frac{1}{2}\frac{\sin (\frac{\pi}{2}k)}{\frac{\pi}{2}k} & \text{otherwise}\end{cases}$

Take a look at a triangle pulse train with a period of T.

This signal is created by circularly convolving the square pulse with itself. The Fourier coefficients for this signal are
${a}_{k}={c}_{k}^{2}=\frac{1}{4}\frac{\sin (\frac{\pi}{2}k)^{2}}{(\frac{\pi}{2}k)^{2}}$

Find the Fourier coefficients of the signal that is created when the square pulse and the triangle pulse are convolved.

If
$f(n)\stackrel{\text{DFT}}{\leftrightarrow}F(k)$ then
$f({((n-m))}_{N})\stackrel{\text{DFT}}{\leftrightarrow}e^{-(i\frac{2\pi}{N}km)}F(k)$ (
i.e. circular shift in time domain =
phase shift in DFT)

other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand

Mian

if the demand increase supply also increases

Thembi

you are wrong this is the law of demand and not the definition

Tarasum

Demand is the willingness of buy and ability to buy in a specific time period in specific place.
Mian you are saying law of demand but not in proper way. you have to keep studying more. because its very basic things in Economics.

Hamza

Demand is the price of Quantity goods and services in which consumer's are willing and able to offer at a price in the market over a period of time

Umar

Demand is the quantity of goods and services that the consumer are willing and able to buy at a alternative prices over a given period of time. But mind you demand is quite different from need and want.

Tarasum

Demand can be defined as the graphical representation between price&demand

alkasim

sorry demand is nt a graphical representation between price and quantity demand but instead that is demand curve.

Ebrima

Demand is the willingness and ability of a consumer to buy a quantity of a good over a given period of time assuming all other things remain constant.

Answer: GPA stands for Grade Point Average. It is a standard way of measuring academic achievement in the U.S. Basically, it goes as follows: Each course is given a certain number of "units" or "credits", depending on the content of the course.

monopsony is when there's only one buyer while monopoly is when there's only one producer.

Bola

who have idea on Banter

Ibrahim

like trade by barter?

Bola

Monopoly is when there's excessively one seller and there is no entry in the market while monopsony is when there is one buyer

kemigisha

Adam smith was born in 1723

Bola

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Good humoured, playful, typically spontaneous conversation.
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Umar

which book Adam smith published first? the first book of Adam smith pls.

Umar

wealth on nation, 1776

Daniel

what is market power and how can it affect an economy?

market power:- where a firm is said to be a price setter.market power benefits the powerful at the expense of others.

Umar

Market power refers to the ability of a firm (or group of firms) to raise and maintain price above the level that would prevail under competition is referred to as market or monopoly power. The exercise of market power leads to reduced output and loss of economic welfare

Kartheek

find information about the national budget

Molahlegi

three branches of economics in which tourism is likely to figure

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

Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.