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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 k x k h n k
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 N and thus the multiplication of these two DFTs will be nonzero for n 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 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 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 . f n g n is the circular convolution of two periodic signals and is equivalent to the convolution over one interval, i.e. f n g n n 0 N η 0 N f η g n η .

Circular convolution in the time domain is equivalent to multiplication of the Fourier coefficients.
This is proved as follows
a k 1 N n 0 N v n j ω 0 k n 1 N 2 n 0 N η 0 N f η g n η ω j 0 k n 1 N η 0 N f η 1 N n 0 N g n η j ω 0 k n ν ν n η 1 N η 0 N f η 1 N ν η N η g ν j ω 0 ν η 1 N η 0 N f η 1 N ν η N η g ν j ω 0 k ν j ω 0 k η 1 N η 0 N f η d k j ω 0 k η d k 1 N η 0 N f η j ω 0 k η c k d k

Circular convolution formula

What happens when we multiply two DFT's together, where Y k is the DFT of y n ?

Y k F k H k
when 0 k N 1

Using the DFT synthesis formula for y n

y n 1 N k 0 N 1 F k H k j 2 N k n

And then applying the analysis formula F k m 0 N 1 f m j 2 N k n

y n 1 N k 0 N 1 m 0 N 1 f m j 2 N k n H k j 2 N k n m 0 N 1 f m 1 N k 0 N 1 H k j 2 N k n m
where we can reduce the second summation found in the above equation into h ( ( n m ) ) N 1 N k 0 N 1 H k j 2 N k n m y n m 0 N 1 f m h ( ( n m ) ) N which equals circular convolution! When we have 0 n N 1 in the above, then we get:
y n f n h n
The notation 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 m 0 N 1 f m h ( ( n m ) ) N For each n : N multiples, N 1 additions

N points implies N 2 multiplications, N N 1 additions implies O N 2 complexity.

Steps for circular convolution

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:

for m -2

To cyclic shift we follow these steps:

1) Write f n on a cylinder, ACW

N 8

2) To cyclic shift by m , spin cylinder m spots ACW f n f (( n + m )) N

m -3

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.

f n
f (( - n )) N

Convolve (n = 4)

Two discrete-time signals to be convolved.

  • h ( ( m ) ) N

Multiply f m and sum to yield: y 0 3

  • h ( ( 1 m ) ) N

Multiply f m and sum to yield: y 1 5

  • h ( ( 2 m ) ) N

Multiply f m and sum to yield: y 2 3

  • h ( ( 3 m ) ) N

Multiply f m and sum to yield: y 3 1

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Exercise

Take a look at a square pulse with a period of T.

For this signal c k 1 N k 0 1 2 2 k 2 k

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 1 4 2 k 2 2 k 2

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

a k = undefined k = 0 1 8 s i n 3 [ π 2 k ] [ π 2 k ] 3 otherwise

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Circular shifts and the dft

Circular shifts and dft

If f n DFT F k then f (( n - m )) N DFT 2 N k m F k ( i.e. circular shift in time domain = phase shift in DFT)

f n 1 N k 0 N 1 F k 2 N k n
so phase shifting the DFT
f n 1 N k 0 N 1 F k 2 N k n 2 N k n 1 N k 0 N 1 F k 2 N k n m f (( n - m )) N

Circular convolution demonstration

circularshiftsDemo
Interact (when online) with a Mathematica CDF demonstrating Circular Shifts.

Conclusion

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

Questions & Answers

List and explain four factors of production
Vuyo Reply
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Thembi
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Ogodo Reply
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Thembi
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akin Reply
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akin Reply
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
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Demand can be defined as the graphical representation between price&demand
alkasim
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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.
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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.
Yusuf
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founder , that is Adam Smith
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The wealth of Nations
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Umar
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Umar
17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
Yusuf
1790
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Umar
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mode is the highest occurring frequency in a distribution
Bola
mode is the most commonly occurring item in a set of data.
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monopsony is when there's only one buyer while monopoly is when there's only one producer.
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like trade by barter?
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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
 (uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
Umar
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Umar
wealth on nation, 1776
Daniel
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Gab Reply
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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
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three branches of economics in which tourism is likely to figure
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Difference between extinct and extici spicies
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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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