Upcoming SlideShare
×

# Final document

1,648
-1

Published on

Published in: Education, Technology
2 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total Views
1,648
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
84
0
Likes
2
Embeds 0
No embeds

No notes for slide

### Final document

1. 1. ABSTRACT <br />Many image processing operations such as scaling and rotation require re-sampling or convolution filtering for each pixel in the image. Convolutions on digital images are important since they represent operations that are more general than the operations that can be performed on analog images. Convolution has many applications which have great significance in discrete signal processing. It is usually difficult to deal with analog signals. Hence signals are converted to digital state. Filtering of signals is very important in order to determine which one to accept and which one to reject, and all of that is done by convolution.<br />This paper presents a direct method of reducing convolution processing time using hardware computing and implementations of discrete linear convolution of two finite length sequences (NXN). This implementation method is realized by simplifying the convolution building blocks. The purpose of this research is to prove the feasibility of an FPGA that performs a convolution on an acquired image in real time. <br />The proposed implementation uses a modified hierarchical design approach, which efficiently and accurately speeds up computation; reduces power, hardware resources, and area significantly. The efficiency of the proposed convolution circuit is tested by embedding it in a top level FPGA. In addition, the presented circuit uses less power consumption and delay from input to output. It also provides the necessary modularity, expandability, and regularity to form different convolutions for any number of bits. <br />CHAPTER 1<br />INTRODUCTION<br />INTRODUCTION <br />Convolution provides the mathematical framework for DSP. It is the single most important technique in Digital Signal Processing. Convolution is a mathematical way of combining two signals to form a third signal. Using the strategy of impulse decomposition, systems are described by a signal called the impulse response. In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change. It has applications that include statistics, computer vision, image and signal processing, electrical engineering, and differential equations.<br />INTRODUCTION TO CONVOLUTION <br /> One of the most important concepts in Fourier theory, and in crystallography, is that of a convolution. Convolutions arise in many guises, as will be shown below. Because of a mathematical property of the Fourier transform, referred to as the convolution theorem, it is convenient to carry out calculations involving convolutions. <br />Convolution Definition<br /> <br />The convolution of ƒ and g is written ƒ∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:<br />      <br /> While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function ƒ(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.<br />More generally, if f and g are complex-valued functions on Rd, then their convolution may be defined as the integral:<br /> TYPES OF CONVOLUTION <br /> There are two types of convolution. They are:<br /><ul><li>Linear convolution
2. 2. Circular convolution
3. 3. Linear convolution</li></ul>Convolution is an integral concatenation of two signals. It has many applications in numerous areas of signal processing. The convolution described above is nothing but linear convolution. The most popular application is the determination of the output signal of a linear time-invariant system by convolving the input signal with the impulse response of the system. Convolving two signals is equivalent to multiplying the Fourier transform of the two signals. <br />1.3.1.1 Mathematical Formulae:<br />The linear convolution of two continuous time signals and is defined by, <br />For discrete time signals x(n) and h(n) , the integration is replaced by a summation <br /><ul><li>Circular Convolution</li></ul>The circular convolution of two aperiodic functions occurs when one of them is convolved in the normal way with a periodic summation of the other function. It occurs naturally in digital signal processing when DTFTs and inverse DTFTs are replaced by DFTs and inverse DFTs. Equivalently, the continuous frequency domain is replaced by a discrete one. (See Circular convolution theorem.)<br />The Circular convolution theorem states that :<br />For a periodic function xT(t) , with period T, the convolution with another function, h(t), is also periodic, and can be expressed in terms of integration over a finite interval as follows:<br />Where, to is an arbitrary parameter, and hT(t) is a periodic summation of h, defined by:<br />When xT(t) is expressed as the periodic summation of another function x, this convolution is sometimes referred to as circular convolution of functions h and x.<br /><ul><li> PROPERTIES OF CONVOLUTION
4. 4. This section describes the properties of convolution. The properties of convolution are:
5. 5. Commutative
6. 6. Associative
7. 7. Distributive
8. 8. Commutative property: